Step 1 The objective of the question is to understand why the probability of the event '3 sets and V wins' is calculated as the sum of two probabilities (0.09 and 0.0675) instead of their product, which is the usual method for calculating the probability of the intersection of two events. Step 2 The key to understanding this lies in the nature of the events in question. In this case, the events '3 sets and V wins' are not independent events, but rather they are mutually exclusive events. Mutually exclusive events are events that cannot occur at the same time. In other words, if one event occurs, the other cannot. Step 3 In the context of this question, the events '3 sets and V wins' are mutually exclusive because they represent different ways in which Player V can win the match. Specifically, 'VMV' and 'MWV' are two different sequences of wins and losses that result in Player V winning the match in 3 sets. Since these sequences cannot occur at the same time, they are mutually exclusive. Step 4 The probability of mutually exclusive events is calculated as the sum of their individual probabilities. This is why we add the probabilities 0.09 and 0.0675 to get the probability of the event "3 sets and V wins'! Step 5 To calculate the conditional probability P(3 sets | V wins), we use the formula P(A|B) = P(A and B) / P(B). Substituting the given values, we get P(3 sets | Vwins) = P(3 sets and V wins) / P(V wins) = (0.09 +0.0675)/ 0.4575=0.1575/0.4575 = 0.344. Solution The probability of the event '3 sets and V wins' is calculated as the sum of the probabilities of the mutually exclusive events 'VMV' and 'MW'. This is because these events represent different ways in which Player V can win the match in 3 sets, and they cannot occur at the same time. The conditional probability P(3 sets | V wins) is then calculated using these probabilities, resulting in a value of approximately 0.344. Part (a): The possible outcomes are listed below, organized by who wins the match. Within each match winner category, who wins each set is shown. 1) 11) Player V wins: vv VMV MVV Player M wins. MM MVM VMM Part (b): The ways in which Player V can win a match against Player M and the corresponding probabilities are shown below. Adding the probabilities for the various ways Player V wins the match yields the overall probability of 0.4575. Outcome VV VMV MVV Probability (0.5)(0.6)=0.3 (0.5)(1 -0.6) (0.45) = 0.09 (0.5)(1 0.7)(0.45) = 0.0675 Total: 0.3 + 0.09 +0.0675 = 0.4575 Part (c): P(3 sets | V wins) = P(3 sets and V wins) (0.09 +0.0675) P(V wins) 0.4575 0.1575 ≈ 0.344 0.4575

Step 1 The objective of the question is to understand why the probability of the event '3 sets and V wins' is calculated as the sum of two probabilities (0.09 and 0.0675) instead of their product, which is the usual method for calculating the probability of the intersection of two events. Step 2 The key to understanding this lies in the nature of the events in question. In this case, the events '3 sets and V wins' are not independent events, but rather they are mutually exclusive events. Mutually exclusive events are events that cannot occur at the same time. In other words, if one event occurs, the other cannot. Step 3 In the context of this question, the events '3 sets and V wins' are mutually exclusive because they represent different ways in which Player V can win the match. Specifically, 'VMV' and 'MWV' are two different sequences of wins and losses that result in Player V winning the match in 3 sets. Since these sequences cannot occur at the same time, they are mutually exclusive. Step 4 The probability of mutually exclusive events is calculated as the sum of their individual probabilities. This is why we add the probabilities 0.09 and 0.0675 to get the probability of the event "3 sets and V wins'! Step 5 To calculate the conditional probability P(3 sets | V wins), we use the formula P(A|B) = P(A and B) / P(B). Substituting the given values, we get P(3 sets | Vwins) = P(3 sets and V wins) / P(V wins) = (0.09 +0.0675)/ 0.4575=0.1575/0.4575 = 0.344. Solution The probability of the event '3 sets and V wins' is calculated as the sum of the probabilities of the mutually exclusive events 'VMV' and 'MW'. This is because these events represent different ways in which Player V can win the match in 3 sets, and they cannot occur at the same time. The conditional probability P(3 sets | V wins) is then calculated using these probabilities, resulting in a value of approximately 0.344. Part (a): The possible outcomes are listed below, organized by who wins the match. Within each match winner category, who wins each set is shown. 1) 11) Player V wins: vv VMV MVV Player M wins. MM MVM VMM Part (b): The ways in which Player V can win a match against Player M and the corresponding probabilities are shown below. Adding the probabilities for the various ways Player V wins the match yields the overall probability of 0.4575. Outcome VV VMV MVV Probability (0.5)(0.6)=0.3 (0.5)(1 -0.6) (0.45) = 0.09 (0.5)(1 0.7)(0.45) = 0.0675 Total: 0.3 + 0.09 +0.0675 = 0.4575 Part (c): P(3 sets | V wins) = P(3 sets and V wins) (0.09 +0.0675) P(V wins) 0.4575 0.1575 ≈ 0.344 0.4575

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter12: Probability

Section12.3: Conditional Probability; Independent Events; Bayes' Theorem

Problem 66E: Working Women A survey has shown that 52 of the women in a certain community work outside the home....

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