A First Course in Probability (10th Edition)
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ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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12.
![**Understanding Binomial Distribution**
In this section, we will explore how to use the binomial probability formula to determine the probability of achieving a specific number of successes in a fixed number of trials.
**Description:**
Given a procedure that yields a binomial distribution, we will focus on a scenario where the trial is repeated \( n \) times. The goal is to determine the probability of obtaining \( x \) successes, given that the probability \( p \) of success on a single trial is known.
To calculate this, we will use the binomial probability formula and round the final answer to three decimal places for precision.
**Formula:**
The binomial probability formula is:
\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
Where:
- \( P(X = x) \): The probability of getting exactly \( x \) successes.
- \(\binom{n}{x}\): The number of combinations of \( n \) items taken \( x \) at a time.
- \( p \): The probability of success on a single trial.
- \( n \): The total number of trials.
- \( x \): The number of successes.
**Example Problem:**
In this example, the values are given as follows:
- \( n = 10 \): Number of trials
- \( x = 2 \): Number of successes
- \( p = \frac{1}{3} \): Probability of success
By substituting these values into the binomial probability formula, students are required to find the probability of exactly 2 successes out of 10 trials. The final answer should be rounded to three decimal places.
**Detailed Steps:**
1. **Calculate the binomial coefficient** \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]
2. **Calculate the term** \( p^x \):
\[ \left( \frac{1}{3} \right)^2 = \frac{1}{9} \]
3. **Calculate the term** \( (1 - p)^{n-x} \):
\[ \left( 1 - \frac{1}{3} \right)^{10-2}](https://content.bartleby.com/qna-images/question/ab0e5d97-7b6b-46cc-bf12-d986af136b0a/27f67691-88be-4105-a419-1fdca658f01a/pvbdlr_processed.jpeg)
Transcribed Image Text:**Understanding Binomial Distribution**
In this section, we will explore how to use the binomial probability formula to determine the probability of achieving a specific number of successes in a fixed number of trials.
**Description:**
Given a procedure that yields a binomial distribution, we will focus on a scenario where the trial is repeated \( n \) times. The goal is to determine the probability of obtaining \( x \) successes, given that the probability \( p \) of success on a single trial is known.
To calculate this, we will use the binomial probability formula and round the final answer to three decimal places for precision.
**Formula:**
The binomial probability formula is:
\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]
Where:
- \( P(X = x) \): The probability of getting exactly \( x \) successes.
- \(\binom{n}{x}\): The number of combinations of \( n \) items taken \( x \) at a time.
- \( p \): The probability of success on a single trial.
- \( n \): The total number of trials.
- \( x \): The number of successes.
**Example Problem:**
In this example, the values are given as follows:
- \( n = 10 \): Number of trials
- \( x = 2 \): Number of successes
- \( p = \frac{1}{3} \): Probability of success
By substituting these values into the binomial probability formula, students are required to find the probability of exactly 2 successes out of 10 trials. The final answer should be rounded to three decimal places.
**Detailed Steps:**
1. **Calculate the binomial coefficient** \( \binom{10}{2} \):
\[ \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45 \]
2. **Calculate the term** \( p^x \):
\[ \left( \frac{1}{3} \right)^2 = \frac{1}{9} \]
3. **Calculate the term** \( (1 - p)^{n-x} \):
\[ \left( 1 - \frac{1}{3} \right)^{10-2}
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