Indeed, the definition of $ \textup{P}$ is to sum over the probabilities of outcomes in an event. hide. But previous slide de nes \probability conditioned on Y = y" and PfY = yg= 0. Given that the sum of the two faces equals to 10, what is the probability Aug 17, 2020 · What is the (conditional) probability that the first turns up six, given that the sum is \(k\), for each \(k\) from two through 12? What is the (conditional) probability that at least one turns up six, given that the sum is \(k\), for each \(k\) from two through 12? Jan 18, 2017 · Conditional probability of two fair dice rolling resulting in sum of 11 and at least one being 5 3 Roll two balanced dice until the sum of the faces equals 7 appears for the first time. Figure 7. Out of $6^3$ possible rolls of three dice, only three (112, 121, 211) sum to 4. It is the probability of the In the conditional probability formula, a division by is performed. In this case, the original sample space can be thought of as a set of 100, 000 females. Recalling that outcomes in this sample space are equally likely, we apply the definition of conditional probability ( Definition 2. P ( D ∩ +) = ‍. 5 Conditional Probability. And the conditional probability, that he eats a bagel for breakfast given that he eats a pizza for lunch, so probability of event A happening, that he eats a bagel for breakfast, given that he's had a pizza for lunch is equal to 0. Jun 25, 2021 · A major hypothesis about conditionals is the Equation in which the probability of a conditional equals the corresponding conditional probability: p(if A then C) = p(C|A). Jul 31, 2023 · Solution. If E is the event that at least one dice lands on 6 and F is the event that the dice land on different numbers, I need to calculate P(EF Apr 23, 2022 · Similarly, we would expect about 28% or 0. , ). We have to count the outcomes all over again. We can pronounce Pr ( A ∣ B) as the probability of event A occurring given that B has occurred. Oct 14, 2019 · Let X X and Y Y be two independent random variables such that X > a X > a and a < X + Y < b a < X + Y < b. P(X1 = 1 ∣ Sn = k) is the probability that a particular trial (the first) is a success when given that exactly k among the n trials are successes. Here are some examples that well describe the process of finding probability. Jul 1, 2020 · The multiplication rule and the addition rule are used for computing the probability of A and B, and the probability of A or B for two given events A, B. We are asked to find the conditional probability that the sum of the numbers on the dice is greater than 7 given that neither die shows a 1. Conditional probability. Look at the numerators in the fractions involved in the sum: the 3 represents the number of E tiles and the 2 is the number of S tiles. When applied to a healthy person, the Examples of Conditional Probability . In sampling with replacement each member has … Apr 1, 2020 · Conditional probability distribution with geometric random variables. The probability of drawing a red ball in the second draw too is an example of conditional probability where the drawing of the second ball depends on the drawing of the first ball. Jul 3, 2024 · Conditional Probability is defined as the probability of any event occurring when another event has already occurred. Solved Example 1: If a fair die is rolled twice, then find the conditional probability that the total of the numbers on the faces is 7, given that the first number is 3. If \( B \subseteq A \) then \( A \) becomes a certain event. 5)$$ This format is particularly useful in situations when we know the conditional probability, but we are interested in the probability of the intersection. , the set of all its possible values, denoted by ): then, we compute the conditional pmf as follows: Statistics and Probability; Statistics and Probability questions and answers; Two distinguishable, fair dice are rolled (one red and one green). I'm a bit confused by this. Where am I going wrong? Apr 16, 2024 · Ex 13. F. We can also study conditional distributions of random variables given the values of some A lot of difficult probability problems involve conditional probability. F). However, if you choose to roll the dice one at a time, the probability of rolling a 10 will change after the first die comes to rest. Jan 5, 2017 · I've attempted a proof of this statement for the discrete (sum) case: Proof: By the Kolmogorov definition of conditional probability and the Law of Total Probability, $$\sum_k P(A_k | B) = \sum_k \frac{P(A_k \cap B)}{P(B)} = \frac{1}{P(B)}\sum_kP(A_k \cap B) = \frac{1}{P(B)}P(B)=1. Conditional variance. I know that the distribution of each Xi ∣ S = s is also Gaussian. That is, the conditional probabilities are between 0 and 1, inclusive: \ (0 \leq g (x|y) \leq 1 \qquad \text {and}\qquad 0 \leq h (y|x) \leq 1 \) and, for each subpopulation, the conditional probabilities sum to 1: According to the sum rule, the probability that any of several mutually exclusive events will occur is equal to the sum of the events’ individual probabilities. The conditional probability distribution is how we measure the probability that a variable takes on some value when we have knowledge about some other variable(s). I Our standard de nition of conditional probability is P(AjB) = P(AB)=P(B). In symbols, if $ E_1, \dots, E_m$ form our partition, then. 1 (Conditional probability) If P(F) >0, we de ne the probability of Egiven Fas P(EjF) := P(E\F) P(F): Note P(E\F) = P Definition (Conditional Probability): the conditional probability of an event A given that an event B has occurred, written Pr ( A ∣ B) is: Pr ( A ∣ B) = Pr ( A ∩ B) Pr ( B). This is an example of a conditional probability. According to clinical trials, the test has t. For independent X and Y random variable which follows distribution Po($\lambda$) and Po($\mu$). align} where (a) holds by definition of conditional probability; (b) holds Now, each of the 36 ordered pairs in the table represent an equally likely outcome. Also, ∑x i=1 x = x2 ∑ i = 1 x x = x 2; you may have gotten confused with ∑x i=1 i ∑ i = 1 x i. Dec 6, 2019 · Probability for a single random variable is straight forward, although it can become complicated when considering two or more variables. Conditional probability: p(A|B) is the probability of event A occurring, given that event B occurs May 12, 2019 · Calculate the conditional probability that the sum of two dice tosses is even given that at least one of the tosses gives a five. This is a sort of implication, but for probabilities. • 2:35 Bob has three coins, two are fair, • 2:43 one is biased, which is weighted to land heads • 2:46 two thirds of the time and tails one third. tain medical condition. It is represented as P (A | B) which means the probability of A when B has already happened. This division is impossible when is a zero-probability event (i. As poisson distribution is a discrete probability distribution, P. ” The marginal probability is different from the conditional probability (described next) because it considers the union of all events for the second variable rather than the probability of a single event. An expectation of a random variable with respect to a regular conditional probability is equal to its conditional expectation. With just two variables, we may be interested in the probability of two simultaneous events, called joint probability: the probability of one event given the occurrence of another event called the conditional probability, or just the probability of an event We would like to show you a description here but the site won’t allow us. Apply the Multiplication Rule for Probability to compute probabilities. Bayes' theorem is named after the Reverend Thomas Bayes ( / beɪz / ), also a statistician and philosopher. We can also do P(X1 = 1 ∣ ∑nj = 1Xj = k) = P(X1 = 1)P( ∑nj = 2Xj = k − 1) P( ∑nj = 1Xj = k) = p My reasoning comes from doing a logic tree where you have $\frac{1}{5}$ probability of choosing 2, followed by $\frac{1}{5}$ probability of choosing 2, followed by $\frac{1}{5}$ probability of choosing 8 (in order to get a sum of 12). Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function. my method is: let Z = X + Y Z = X + Y and X1 = X X 1 = X ,after finding the joint pdf of Z Z and X X, fXZ(x, z) = λ2x+2y×e− Another important method for calculating conditional probabilities is given by Bayes's formula. • 2: Another important method for calculating conditional probabilities is given by Bayes's formula. Property 1: Let E and F be events of a sample space S of an experiment, then we have P(S|F) = P(F|F) = 1. . For example, if you roll a six-sided die, you have a 1 / 6 ‍ chance of getting any given number, but you can only get one number per roll. Dependent and independent events. 29). A fair die is rolled, Let A be the event that shows an outcome is an odd number, so A={1, 3, 5}. Explanation: Sure, let's solve this step by step. 1, 10 A black and a red dice are rolled. Note that the above equation simply describes how to go from a joint probability mass function P(x, y) P ( x, y) to the probability mass function P(x) P ( x) (or P(y) P ( y) ), that is, by summing out the Even some of the outcomes that give the sum of 7 or 9 are impossible. 75 ‍. an integer, like 6 ‍. Bayes used conditional probability to provide an algorithm (his Proposition 9) that uses evidence to calculate limits on an unknown parameter. But, straight from the definition of conditional expectation, it isn't clear that symmetry in the joint distributions is enough to get the result. We’ll recap some basic probability rules, look at mutually exclusive or disjoint events, play with Venn diagrams, and learn how to work out whether two events are independent. User Did's comment points out that the symmetry comes from the fact that $(\xi, \eta)$ and $(\eta, \xi)$ are identically distributed. The probability of event A and event B occurring together. In this video, we’re going to learn about conditional probability. 280 470 = 0. Jan 8, 2023 · The Markov Assumption above is a conditional probability distribution. Conditional Probabilitypharmaceutical company is marketing a new test for a ce. Feb 3, 2017 · 1. The product rule just shows you how you convert a conditional probability to a joint probability. $\begingroup$ Did-Thx. Example: the probability that a card is a four and red =p(four and red) = 2/52=1/26. • 2:26 In fact, all conditional probability questions • 2:29 can be solved by growing trees. It is. We For example, 3 of these 36 equally likely outcomes correspond to rolling a sum of 10, so the probability of rolling a 10 is 3 36 = 1 12 3 36 = 1 12. 1. Answer: The conditional probability that the sum is greater than 7 given that neither die is a one is 36/125. (1) We represent probabilities on the Calculate conditional probabilities. 1 3. Joint probability of sum of iid random variables and components. A conditional probability can be computed relative to a probability measure that is itself a conditional probability Aug 17, 2020 · In addition to its properties as a probability measure, conditional probability has special properties which are consequences of the way it is related to the original probability measure \(P(\cdot)\). Back in Example 7. Jun 23, 2023 · We are asked to find the probability that a sum of 5 is obtained after learning that the first dice landed on a "3". Sep 6, 2019 · However, every resource I can find on joint probability distributions shows them as a product of two other distributions, not their sum. If we want to be able to define also when , then we need to give a more complicated definition of conditional probability. Symmetry of the situation should immediately suggests this probability is k / n. Let’s shade those in (Figure 7. 2. Conditional probability; Product rule; Independence; Product rule for independent events . 5. I could probably work out analogous The conditional probability that the second card is an Ace given that the first card is an Ace is thus 0. Compare with the conditional probability density function in the previous exercise. The formula is based on the expression P(B) = P(B|A)P(A) + P(B|A c)P(A c), which simply states that the probability of event B is the sum of the conditional probabilities of event B given that event A has or has not occurred. 2. a mixed number, like 1 3 / 4 ‍. So let me write this down. 28. 9%. A bag contains 3 blue balls and 7 red balls. Step 3: To find probability, divide n (A) by n (S). In probability theory, the law of total covariance, [1] covariance decomposition formula, or conditional covariance formula states that if X, Y, and Z are random variables on the same probability space, and the covariance of X and Y is finite, then. Conditional Probability Dec 8, 2019 · The "probability" of the upper triangle matrix is $\left(1-(sum\:of\:the\:diagonal\:elements)\right)/2$ Find conditional distribution of a sum of two random What is conditional probability and how does it relate to independence? Learn how to use formulas and tables to calculate conditional probabilities and check if two events are independent. 0. 13. What is the conditional probability that at least one lands on 6 given that the dice land on different numbers? I already know the answer, but am having some trouble understanding it. The derivation involves two steps: first, we compute the marginal probability mass function of by summing the joint probability mass over the support of (i. Step 3: Since the event we’re interested in is the one consisting of rolls of 4, 5, or 7. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a Jun 29, 2021 · I don't really understand how to apply the sum rule of probability to get this result. Conditional Probability Properties. Find the following probabilities: The probability that the second card is a heart given that the first card is a spade. In this section, let’s understand the concept of conditional probability with some easy examples; Example 1 . • 2:32 Let's do one more to be sure. All these examples of conditional probability have one thing in common: we assume that something is known before calculating a probability. The probability that the first card is a face card and the Jul 17, 2019 · 3. Bayes' theorem is a formula that describes how to update the probabilities of hypotheses when given evidence. Example 1: Find the probability of getting a number less than 5 when a dice is rolled by using the probability formula. Apr 24, 2022 · Parts (a) and (c) certainly make sense. Let S be their sum. Mentor: Mathematicians would say that our question is about conditional probability, because it asks: "What is the probability of Event A on condition of Event B? That is the same thing as "in the case of Event B. Conditional distributions. Firstly, though, let’s recall some probability rules. The probability of A conditioned on B, denoted P(A|B), is equal to P(AB)/P(B). If \( A \cap B = \emptyset \) then \( A \) becomes an impossible event. Given a hypothesis H H and evidence E E, Bayes' theorem states that Jul 18, 2022 · Example 3. It expresses the total probability of an outcome which can be realized via several distinct events, hence the name. Oct 6, 2016 · $\Bbb P(N=2)=\frac14$ and you have already calculated that only three rolls of the 36 possible with two dice sum to 4, so $\Bbb P(N=2\cap S=4)=\frac14×\frac1{12}=\frac1{48}$. " Chain rule for conditional probability: Let us write the formula for conditional probability in the following format $$\hspace{100pt} P(A \cap B)=P(A)P(B|A)=P(B)P(A|B) \hspace{100pt} (1. For example, if the chance of A happening is 50%, and the same for B, what are the chances of both happening, only one happening, at least one happening, or neither happening, and so on. 7, which is interesting. If A and B are said to be mutually exclusive events then the probability of an event A occurring or the probability of event B occurring that is P (a ∪ b) formula is given by P(A) + P(B), i. In probability theory and statistics, a conditional variance is the variance of a random variable given the value (s) of one or more other variables. Khan Academy is a free online learning platform that covers various topics in math, science, and more. 1 ) and find The probability of the intersection of A and B may be written p(A ∩ B). 5%/7. Example of independent events: dice and coin 5 days ago · With the probability calculator, you can investigate the relationships of likelihood between two separate events. What you can write however is. a simplified proper fraction, like 3 / 5 ‍. De nition 4. Find the probability that the chosen cards are odd-numbered. A conditional probability is regular if \operatorname {P} (\cdot|\mathcal {B}) (\omega) P(⋅∣B)(ω) is also a probability measure for all \omega ∈ \Omega ω ∈ Ω. Since both dies are rolled S = We need to find the Probability of obtaining a sum greater than 9, given that the black die r In probability theory, the law (or formula) of total probability is a fundamental rule relating marginal probabilities to conditional probabilities. 7% = 5. In Probability, Bayes theorem is a mathematical formula, which is used to determine the conditional probability of the given event. My understanding of conditional probability in the case of continuous random variables is that P[Y lies in Borel set A|X=x] =Integral over A of the density f(x,y)/g(x); f is the joint density of x,y and g is the marginal density of x. Not sure I understand the math behind the identity, though the intuition is clear as it's similar to discrete case. • 2:50 He chooses a coin at random and flips it. Conditional probability refers to the probability of an event given that another event occurred. P(A or B) = P(A) + P(B) Let’s use this addition rule to find the probability for Experiment 1. Let E represent the event that the red die is a 2. In other words, it calculates the probability of one event happening given that a certain condition is satisfied. $\endgroup$ – elfeck Commented Jun 3, 2014 at 10:27 Oct 30, 2017 · Yes, although the case with Y = N Y = N is really boring: you are taking the sum of N N copies of E[N|N] E [ N | N], so it's almost trivial that the answer is N2 N 2. Please how do I simplify further, the conditional probability Pr(a < X + Y < b∣∣X > a) Pr ( a < X + Y < b | X > a) ? I am guessing that one of the final terms will involve convolution of the sum X + Y X + Y, but I don't know how to go Aug 30, 2017 · 0. In this theory, intuitive models (system 1) do not represent what is false, and so Notice that the probability of drawing an E is 3 10 3 10 and the probability of drawing an S is 2 10 2 10; adding those together, we get 3 10 + 2 10 = 5 10 3 10 + 2 10 = 5 10. First, it is important to distinguish between dependent and independent events! The intuition is a bit different in both cases. With this in mind, we give the following de nition. Like any probability distribution: Probability cannot be negative; The probabilities must sum to 1; The Mar 28, 2013 · Our first observation is quite a trivial one: the probabilities of the events in a partition sum to one. The proposition in probability theory known as the law of total expectation, [1] the law of iterated expectations [2] ( LIE ), Adam's law, [3] the tower rule, [4] and the smoothing theorem, [5] among other names, states that if is a random variable whose expected value is defined, and is any random variable on the same By deriving the conditional probability mass function of . 1 Conditional Probability for Drawing Cards without Replacement. I Doesn’t make sense if P(B) = 0. Feb 6, 2021 · Let's calculate the conditional probability of \(A\) given \(D\), i. Let F represent the event that the sum of the two dice is 7? Find the conditional probability, P(E | F), and express your answer as a fraction in May 17, 2018 · Let X = (X1, X2, …, Xn) be jointly Gaussian with mean vector μ and covariance matrix Σ. Two cards are drawn from a well shuffled deck of 52 cards without replacement. Ok. , the probability that at least one heads is recorded (event \(A\)) assuming that at least one tails is recorded (event \(D\)). The events \(E\) and \(F\) are the subsets of the sample space consisting of all women who live at least 60 years, and at least 80 years, respectively. The probability of their union is the sum . contributed. It follows simply from the axioms of conditional probability, but can be used to powerfully reason about a wide range of problems involving belief updates. Hence Conditional probability of B on A will be, P(B|A) = 19/29. Suppose that we know that event \( B \) has occurred. 38) to help us find the probabilities associated with rolling two standard 6-sided dice: Figure 7. P(x) is the probability of the vector having its exact configuration (out of all possible finite configurations). Fares selects 2 balls without replacement and draws the following tree diagram. Henry's answer has the essential idea, which is to use symmetry. The following are easily derived from the definition of conditional probability and basic properties of the prior probability measure, and prove The sum rule just says that if you've sliced up the probability of X according to which Y it occurs with, then to reconstitute the probability of X, just add up the probability of the slices. Jan 3, 2024 · Let us take some of the conditional probability questions. $\Bbb P(N=3)=\frac18$. I When can we (somehow) make sense of conditioning on probability zero event? I Tough question in general. When n = 2, I know that E(X1 ∣ S = s) = s σ21 σ21 + σ22 and V(X1 ∣ S = s) = σ21σ22 σ21 + σ22 (see here and here ). \ \square$$ Is this a correct proof? Aug 10, 2022 · An insurance company uses conditional probability when setting rates for car insurance. What is the conditional probability that the first die is six given that the sum of the dice is seven? Conditional probability The possibility of an event happening contingent on the occurrence of a prior event is known as conditional probability. led false negatives ). The division provides that the probabilities of all outcomes within B will sum to 1. Experiment 1: A single 6-sided die is rolled. May 6, 2020 · This is another important foundational rule in probability, referred to as the “sum rule. The nomenclature in this article's title parallels the phrase law of total variance. Two standard dice with 6 sides are thrown and the faces are recorded. i. Question 1: Ten numbered cards are there from 1 to 15, and two cards a chosen at random such that the sum of the numbers on both the cards is even. fits better in this case. e following properties:When applied to an affected person, the test comes up positive in 90% of cases, and negative in 10% (these are c. Find the probability that a randomly selected patient has the disease AND tests positive. In the conditional probability formula, a division by is performed. Let X and Y be independent random variable each Poisson distributed with common parameter λ λ. Find the conditional probability. For a trivial sigma algebra. 3 days ago · Example 1: Finding a Conditional Probability on a Tree Diagram. These can be tackled using tools like Bayes' Theorem, the principle of inclusion and exclusion, and the notion of independence. $\Bbb P(N=3\cap S=4)=\frac18×\frac3{216}=\frac1{576}$. Your answer should be. which is the same as the probability that a person chosen at random is a woman and a smoker divided by the probability that a person chosen at random is a woman. Similarly, the probability of occurrence of B when A has already occurred is given by, P(B|A) = P(B ∩ A)/P(A) To have a better insight let us practice some conditional probability examples. What is the probability of rolling a 2 or a 5? It states that the probability of either event occurring is the sum of probabilities of each event occurring. Apr 21, 2017 · The key lies in reiterating the definition of probability as $$\frac{\rm favorable {\ } cases}{\rm possible {\ } cases}$$ The conditional probabilities are computed with less possible cases. Conditional probability is defined as the likelihood that an event will occur, based on the occurrence of a previous outcome. There are three permutations so I end up with my solution above. Thus, the conditional probability could be computed: P(student = uses | parents = used) = # times student = uses given parents = used # times parents = used. His work was published in 1763 as An Essay Towards Solving a Problem in the Doctrine of Chances. Conditional Probability Questions with Solutions. G. In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. , P (A Or B) = P(A) + P(B) Apr 23, 2022 · Run the simulation 100 times and compute the empirical conditional probability density function of \(X\) given \(Y = 2\). P ( Y = y | X + Y = z) for y = 0, 1,, z. P(Y = y|X + Y = z) for y = 0, 1,, z. 28X1000 = 280 to meet both the information criterion and represent our outcome of interest. The conditional probability of the remaining 30 combinations is 0 since the first die is not a 2 in these cases. By multiplication rule of probability, Conditional distributions are valid probability mass functions in their own right. e. 60. Aug 30, 2018 · Joint probability is the likelihood of more than one event occurring at the same time P (A and B). an exact decimal, like 0. Step 2: To make our analysis easier, let’s replace each ordered pair with the sum (Figure 7. P(a|b) =∑z P(a, z|b), P ( a | b) = ∑ z P ( a, z | b), which is sometimes referred to as marginalization. Probabilistic theories often treat it as axiomatic, whereas it follows from the meanings of conditionals in the theory of mental models. Also, suppose B the event that shows the outcome is less than or equal to 3, so B= {1, 2, 3}. Solution. (a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5. [1] Conditional variances are important parts of The probability of event B, that he eats a pizza for lunch, is 0. To answer this question, we will let \(A = \{ \text{the sum is a 5}\} \) and we will let \(B = \{ \text{the first dice is a 3}\} \). By our definition of conditional probability, we know that Now, only 19 red balls and 10 blue balls are left in the bag. Related. 3. 28). The joint distribution of random variables X X and Y Y (defined on the same probability space) is a probability distribution on (x,y) ( x, y) pairs, and describes how the values of X X and Y Y vary together or jointly. a simplified improper fraction, like 7 / 4 ‍. Can the sum of two conditional probability distributions generally produce a joint probability distribution, or is it some quirky feature of this particular conditional probability distribution?? This is the essence of conditional probability. Oct 25, 2015 · The probability is P(A//B)=1/18 If we denote: A - the sum of 2 dice is 12 B - the sum of 2 dice is even Then we are looking for a conditional probability P(A//B Law of total expectation. 38. For example, say we roll two dice, one after the other (so we can differentiate them), and consider the following statements: Apr 1, 2017 · Probability = 1/6 Let A be the event that the sum of the two dice is 5 Let B be the event that the green die is either a 3 or 4 Then we want P( A | B) which we calculate using the conditional probability formula: P( A | B) = (P(A nn B)) / (P(B)) Consider first P(A nn B) which we can calculate using P(A nn B) = (n(A nn B)) / (n(T)) Where, n(T) is the total number of possible outcomes. 47 = 0. Now, lets look at your integral: p(x = 1 | D) = ∫10p(x Henry's answer has the essential idea, which is to use symmetry. Shouldn't the probability just be 1/2, since we know that at least one of the dice tosses gave us a five, thus the other must give us an odd number? Addition Rule 1: When two events, A and B, are mutually exclusive, the probability that A or B will occur is the sum of the probability of each event. Mar 11, 2023 · Therefore, the conditional probability of the outcomes above is 1/6. Step 1 : Understand the problem. I Consider conditional law of X given that Y 2(y Feb 26, 2015 · Two fair dice are rolled. 28 0. (There are two red fours in a deck of 52, the 4 of hearts and the 4 of diamonds). For example, the insurance company may believe the chance you have an accident is higher if you are younger than 27. Finally, since only one of these six outcomes can sum up to 7, (2,5), the probability is 1/6 for rolling a sum of 7 given the value of the first die is a 2. Solution: Let us obtain the sample space of rolling a die twice. Our probability calculator gives you six Step 1. Let us solve some questions based on conditional probability with detailed solutions. , P (A) = n (A)/n (S). Apr 9, 2020 · Find the probability density function of sum of two marginal probability density functions 1 Joint Probability Mass Function/ Marginal Probability Mass Function You can use Probability Generating Function(P. We will return to this point later. $$\displaystyle \sum_{i=1}^m \textup{P}(E_i) = 1$$. Given that the first ball is red, find the value of 𝑥 that represents the probability that the second ball selected is red. 18, we constructed the following table (Figure 7. gd de vx zj af bt ic sj ym gh