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Chapter 5: Problem 12

In a baseball game, Tommy gets a hit \(30 \%\) of the time when facing this pitcher. Joey gets a hit \(25 \%\) of the time. They are both coming up to bat this inning. a. What is the probability that Joey or Tommy will get a hit? b. What is the probability that neither player gets a hit? c. What is the probability that they both get a hit?

Short Answer

Expert verified
a. 0.475 b. 0.525 c. 0.075

Step by step solution

01

Understanding Individual Probabilities

Tommy's probability of getting a hit is 0.30 and Joey's probability of getting a hit is 0.25.
02

Calculate Probability of Either Joey or Tommy Getting a Hit

Use the formula for the probability of either event happening: \( P(A \cup B) = P(A) + P(B) - P(A \cap B) \) where \( P(A \cap B) \) is the probability that both happen. \( P(T) = 0.30 \), \( P(J) = 0.25 \). Substitute these into the formula: \( 0.30 + 0.25 - (0.30 \times 0.25) = 0.30 + 0.25 - 0.075 = 0.475 \). The probability of either Joey or Tommy getting a hit is 0.475.
03

Calculate Probability that Neither Player Gets a Hit

This is the complement of at least one of them getting a hit. Since the probability of either getting a hit is 0.475, the probability that neither gets a hit is \(1 - 0.475 = 0.525\).
04

Calculate Probability that Both Players Get a Hit

The probability that both get a hit is the product of their individual probabilities: \( P(T \cap J) = P(T) \times P(J) = 0.30 \times 0.25 = 0.075 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complement Rule
The complement rule is a useful concept in probability. It allows you to calculate the probability of an event not happening by subtracting the probability of the event happening from 1. This makes it easy to find the probability of the opposite scenario. In the baseball example, we are interested in finding out the probability that neither Tommy nor Joey gets a hit.
To use the complement rule here, we first found the probability that either Tommy or Joey does get a hit, which is 0.475. The complement of this scenario, meaning neither of them get a hit, is simply:
  • Subtracting 0.475 (probability of either getting a hit) from 1, gives us 0.525.
This tells us that there is a 52.5% chance that neither player makes a hit when facing the pitcher.
Intersection Probability
Intersection probability deals with the likelihood of two events happening at the same time. In other words, it calculates the probability that both events occur simultaneously. For instance, in our baseball example, this would mean both Tommy and Joey hitting the ball in a single inning.
To determine this, you multiply the probabilities of each individual event happening:
  • Tommy's probability of hitting is 0.30.
  • Joey's probability of hitting is 0.25.
So, the probability of both of them hitting is given by the formula:\[P(T \cap J) = P(T) \times P(J) = 0.30 \times 0.25 = 0.075\]This calculation shows us that there is a 7.5% chance that both Tommy and Joey will get a hit when they are up against the pitcher this inning.
Union Probability
Union probability is about finding the chance of either one or another event happening, or both. In probability terms, the word 'union' signifies the occurrence of any one, or more, of several events. In our case, we want to find the probability that either Tommy or Joey, or both, get a hit.
We use the union probability formula here, which is:
  • \[P(A \cup B) = P(A) + P(B) - P(A \cap B)\]
Where
  • \(P(A)\) is the probability of Tommy getting a hit (0.30),
  • \(P(B)\) is the probability of Joey getting a hit (0.25),
  • \(P(A \cap B)\) is the probability that both get a hit (0.075).
By substituting these values in, we calculated:\[0.30 + 0.25 - 0.075 = 0.475\]This means there's a 47.5% chance that either Tommy or Joey gets a hit in this inning. This formula helps avoid over-counting the situation where both could happen simultaneously.

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Most popular questions from this chapter

The following questions are from ARTIST (reproduced with permission) Consider the question of whether the home team wins more than half of its games in the National Basketball Association. Suppose that you study a simple random sample of 80 professional basketball games and find that 52 of them are won by the home team. a. Assuming that there is no home court advantage and that the home team therefore wins \(50 \%\) of its games in the long run, determine the probability that the home team would win \(65 \%\) or more of its games in a simple random sample of 80 games. b. Does the sample information (that 52 of a random sample of 80 games are won by the home team) provide strong evidence that the home team wins more than half of its games in the long run? Explain.

The following questions are from ARTIST (reproduced with permission) You are to participate in an exam for which you had no chance to study, and for that reason cannot do anything but guess for each question (all questions being of the multiple choice type, so the chance of guessing the correct answer for each question is \(1 / \mathrm{d}, \mathrm{d}\) being the number of options (distractors) per question; so in case of a 4 -choice question, your guess chance is 0.25 ). Your instructor offers you the opportunity to choose amongst the following exam formats: I. 6 questions of the 4 -choice type; you pass when 5 or more answers are correct; II. 5 questions of the 5 -choice type; you pass when 4 or more answers are correct; III. 4 questions of the 10 -choice type; you pass when 3 or more answers are correct. Rank the three exam formats according to their attractiveness. It should be clear that the format with the highest probability to pass is the most attractive format. Which would you choose and why?

The following questions are from ARTIST (reproduced with permission) One of the items on the student survey for an introductory statistics course was "Rate your intelligence on a scale of 1 to \(10 . "\) The distribution of this variable for the 100 women in the class is presented below. What is the probability of randomly selecting a woman from the class who has an intelligence rating that is LESS than seven (7)? $$ \begin{array}{|c|c|} \hline \text { Intelligence Rating } & \text { Count } \\ \hline 5 & 12 \\ \hline 6 & 24 \\ \hline 7 & 38 \\ \hline 8 & 23 \\ \hline 9 & 2 \\ \hline 10 & 1 \\ \hline \end{array} $$ a. \((12+24) / 100=.36\) b. \((12+24+38) / 100=.74\) c. \(38 / 100=.38\) d. \((23+2+1) / 100=.26\) e. None of the above.

True/False: You are more likely to get a pattern of HTHHHTHTTH than HHHHHHHHTT when you flip a coin 10 times.

The following questions are from ARTIST (reproduced with permission) A bowl has 100 wrapped hard candies in it. 20 are yellow, 50 are red, and 30 are blue. They are well mixed up in the bowl. Jenny pulls out a handful of 10 candies, counts the number of reds, and tells her teacher. The teacher writes the number of red candies on a list. Then, Jenny puts the candies back into the bowl, and mixes them all up again. Four of Jenny's classmates, Jack, Julie, Jason, and Jerry do the same thing. They each pick ten candies, count the reds, and the teacher writes down the number of reds. Then they put the candies back and mix them up again each time. The teacher's list for the number of reds is most likely to be (please select one): a. 8,9,7,10,9 b. 3,7,5,8,5 c. 5,5,5,5,5 d. 2,4,3,4,3 e. 3,0,9,2,8
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