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Chapter 5: Problem 11

You win a game if you roll a die and get a 2 or a 5 . You play this game 60 times. a. What is the probability that you win between 5 and 10 times (inclusive)? b. What is the probability that you will win the game at least 15 times? c. What is the probability that you will win the game at least 40 times? d. What is the most likely number of wins. e. What is the probability of obtaining the number of wins in d? Explain how you got each answer or show your work.

Short Answer

Expert verified
a. Use binomial distribution to find \( P(5 \leq X \leq 10) \). b. Calculate \( P(X \geq 15) \). c. Compute \( P(X \geq 40) \). d. Most likely wins: 20. e. Probability for 20: \( \binom{60}{20} \left( \frac{1}{3} \right)^{20} \left( \frac{2}{3} \right)^{40} \).

Step by step solution

01

Understanding the Problem

Firstly, we need to understand the probability of winning in a single attempt. Since you win if you roll a 2 or a 5 on a 6-sided die, there are 2 winning outcomes. The probability of winning in one roll is \( \frac{2}{6} = \frac{1}{3} \).
02

Define the Distribution

Since each roll of the die is an independent event and there is a fixed probability of winning, the total number of times you win out of 60 attempts follows a binomial distribution. We have \( n = 60 \) and \( p = \frac{1}{3} \).
03

Winning Between 5 and 10 Times

To find the probability of winning between 5 and 10 times inclusively, calculate \( P(5 \leq X \leq 10) \) with \( X \sim B(60, \frac{1}{3}) \). Use the binomial probability formula: \( P(X = k) = \binom{60}{k} \left( \frac{1}{3} \right)^k \left( \frac{2}{3} \right)^{60-k} \) for \( k = 5 \) to \( 10 \), and sum these probabilities.
04

Winning at Least 15 Times

For the probability of winning at least 15 times, calculate \( P(X \geq 15) = 1 - P(X \leq 14) \). Use cumulative binomial probability distributions to find \( P(X \leq 14) \).
05

Winning at Least 40 Times

Similarly, calculate \( P(X \geq 40) = 1 - P(X \leq 39) \). Use the cumulative binomial probability function to evaluate this.
06

Find the Most Likely Number of Wins

The most likely number of wins corresponds to finding the mode of the distribution, which is around the expected value. Calculate the expected value: \( E(X) = n \cdot p = 60 \times \frac{1}{3} = 20 \). The mode of a binomial distribution is usually near its mean.
07

Probability of the Most Likely Number of Wins

Using the binomial probability formula, compute \( P(X = 20) = \binom{60}{20} \left( \frac{1}{3} \right)^{20} \left( \frac{2}{3} \right)^{40} \) to find the probability of winning exactly 20 times.

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Key Concepts

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

Probability
Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In this particular game, probability helps us understand the chances of winning when rolling a die.

Since there are 6 faces on a die and you win if you roll a 2 or a 5, there are 2 successful outcomes. Probability is calculated by dividing the number of successful outcomes by the total number of possible outcomes.
  • The probability of rolling a 2 or a 5 is \( \frac{2}{6} = \frac{1}{3} \).
Probability in this context helps determine the likelihood of different winning outcomes when the game is played several times.
Binomial Probability Formula
The binomial probability formula is used when there are a fixed number of independent trials, each with the same probability of success. This is exactly the case with our game, where you roll the die 60 times. The formula is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]Where:
  • \( n \) is the number of trials (in this case, 60 rolls).
  • \( k \) is the number of successful trials (wins).
  • \( p \) is the probability of success on a single trial (\( \frac{1}{3} \)).
This formula helps calculate the probability of achieving exactly \( k \) wins. In the problem, it was used to find probabilities for situations like winning between 5 and 10 times or at least 15 times, by summing or subtracting various \( P(X = k) \) values.
Expected Value
The expected value is a key concept in probability that tells you the average outcome if an experiment is repeated a large number of times.

For a binomial distribution, the expected value \( E(X) \) is calculated using the formula:\[ E(X) = n \cdot p \]Here:
  • \( n = 60 \) (the number of times you play)
  • \( p = \frac{1}{3} \) (the probability of winning once)
Thus, the expected number of wins is \( E(X) = 60 \times \frac{1}{3} = 20 \).
This means if you played this game many times, you'd expect to win 20 times on average in 60 plays.
Mode
The mode of a distribution is the most frequently occurring value. In a binomial distribution, the mode can be found around the expected value, especially for larger numbers of trials. For this problem, you need to identify the most likely number of wins.

Theoretically, when n is large, the mode is close to the expected value, which we've calculated as 20 in this situation. However, the mode for a binomial distribution might slightly vary due to the exact arrangement of probabilities. Here:
  • The mode, or most likely number of wins, is 20 based on the symmetry and peak position near the mean value in this distribution.
Using the binomial probability formula, this calculated probability for exactly 20 wins provides the likelihood of achieving this most common outcome.

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Most popular questions from this chapter

The following questions are from ARTIST (reproduced with permission) A refrigerator contains 6 apples, 5 oranges, 10 bananas, 3 pears, 7 peaches, 11 plums, and 2 mangos. a. Imagine you stick your hand in this refrigerator and pull out a piece of fruit at random. What is the probability that you will pull out a pear? b. Imagine now that you put your hand in the refrigerator and pull out a piece of fruit. You decide you do not want to eat that fruit so you put it back into the refrigerator and pull out another piece of fruit. What is the probability that the first piece of fruit you pull out is a banana and the second piece you pull out is an apple? c. What is the probability that you stick your hand in the refrigerator one time and pull out a mango or an orange?

An unfair coin has a probability of coming up heads of \(0.65 .\) The coin is flipped 50 times. What is the probability it will come up heads 25 or fewer times? (Give answer to at least 3 decimal places).

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?

A test correctly identifies a disease in \(95 \%\) of people who have it. It correctly identifies no disease in \(94 \%\) of people who do not have it. In the population, \(3 \%\) of the people have the disease. What is the probability that you have the disease if you tested positive?

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.
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