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Magazine Chapter 6: Problem 19
A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) successes in the \(n\) independent trials of the experiment. $$ n=40, p=0.99, x=38 $$
Short Answer
Expert verified
0.0521
Step by step solution
01
- Understand the Binomial Probability Formula
The binomial probability formula is given by: \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]where \( \binom{n}{x} \) is the binomial coefficient, \( p \) is the probability of success in a single trial, \( n \) is the number of trials, and \( x \) is the number of successes.
02
- Calculate the Binomial Coefficient
Calculate the binomial coefficient \( \binom{40}{38} \) using the formula: \[ \binom{n}{x} = \frac{n!}{x! (n-x)!} \]For our values, \[ \binom{40}{38} = \frac{40!}{38! (40-38)!} = \frac{40!}{38! \times 2!} = \frac{40 \times 39}{2 \times 1} = 780 \]
03
- Calculate the Probability of Success
Calculate the probability term for successes \( p^x \), where \( p = 0.99 \) and \( x = 38 \): \[ 0.99^{38} \]
04
- Calculate the Probability of Failures
Calculate the probability term for failures \( (1-p)^{n-x} \), where \( p = 0.99 \), \( n = 40 \), and \( n - x = 2 \): \[ (1-0.99)^{40-38} = 0.01^{2} \]
05
- Combine All Terms to Find the Probability
Combine all the terms: \[ P(X = 38) = \binom{40}{38} \times 0.99^{38} \times 0.01^2 \]Thus, \[ P(X = 38) = 780 \times 0.99^{38} \times 0.01^2 \]
06
- Compute the Final Answer
Compute the value using a calculator:\[ 780 \times (0.99^{38}) \times (0.01^2) \approx 780 \times 0.6676 \times 0.0001 \approx 0.0521 \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Coefficient
The binomial coefficient, often represented as \( \binom{n}{x} \) or 'n choose x', is a key part of the binomial probability formula.
It calculates the number of ways to choose \( x \) successes out of \( n \) trials. The formula is: \[ \binom{n}{x} = \frac{n!}{x! (n-x)!} \]
Here, \( n \) is the total number of trials, and \( x \) is the number of successes. The exclamation mark '!' denotes factorial, which means multiplying a sequence of descending natural numbers. For instance, \( 5! = 5\times4\times3\times2\times1 \).
This coefficient helps in finding out how many possible ways you can achieve your set number of successes in the given number of trials. In our example, for 40 trials and 38 successes, the binomial coefficient is 780.
It calculates the number of ways to choose \( x \) successes out of \( n \) trials. The formula is: \[ \binom{n}{x} = \frac{n!}{x! (n-x)!} \]
Here, \( n \) is the total number of trials, and \( x \) is the number of successes. The exclamation mark '!' denotes factorial, which means multiplying a sequence of descending natural numbers. For instance, \( 5! = 5\times4\times3\times2\times1 \).
This coefficient helps in finding out how many possible ways you can achieve your set number of successes in the given number of trials. In our example, for 40 trials and 38 successes, the binomial coefficient is 780.
Probability of Successes
In each trial of a binomial experiment, there's a probability of success, denoted by \( p \).
To calculate the probability of having exactly \( x \) successes, you need to raise the probability of success \( p \) to the power of those \( x \) successes. This is given as \( p^x \).
For example, if the probability of success in a single trial is 0.99 and you are looking for 38 successes, then you calculate \( 0.99^{38} \). This step accounts for the likelihood of all the successes occurring as intended in the trials considered.
The calculation for our problem would be finding 0.99 raised to the power of 38. The result of which is 0.6676.
To calculate the probability of having exactly \( x \) successes, you need to raise the probability of success \( p \) to the power of those \( x \) successes. This is given as \( p^x \).
For example, if the probability of success in a single trial is 0.99 and you are looking for 38 successes, then you calculate \( 0.99^{38} \). This step accounts for the likelihood of all the successes occurring as intended in the trials considered.
The calculation for our problem would be finding 0.99 raised to the power of 38. The result of which is 0.6676.
Independent Trials
In a binomial experiment, trials are independent.
This means the outcome of one trial does not affect the outcome of another. Each trial is separate from the others, and the probability of success remains constant across all trials.
For example, when calculating the probability of getting 38 successes out of 40 trials, each trial has no impact on the next. The probability remains the same, \( p = 0.99 \), throughout all 40 trials.
It is this independence that allows the multiplication of the probabilities of successes and failures as separate terms in the binomial probability formula.
This means the outcome of one trial does not affect the outcome of another. Each trial is separate from the others, and the probability of success remains constant across all trials.
For example, when calculating the probability of getting 38 successes out of 40 trials, each trial has no impact on the next. The probability remains the same, \( p = 0.99 \), throughout all 40 trials.
It is this independence that allows the multiplication of the probabilities of successes and failures as separate terms in the binomial probability formula.
Probability of Failures
Just as with successes, there's a probability of failures, denoted as \( 1-p \).
To find the combined probability of the failures occurring across the trials, you raise \( 1-p \) to the power of the number of failures, \( n - x \).
In this example, if \( p = 0.99 \), then \( 1-p = 0.01 \). The number of failures in 40 trials with 38 successes is 2, calculated as \( 40 - 38 = 2 \). Therefore, the probability of failures is given by \[ (1-0.99)^2 = 0.01^2 \].
The result shows the probability of encountering 2 failures in these trials, which helps in computing the overall probability sought. The calculation results in \(0.01^2 = 0.0001\).
To find the combined probability of the failures occurring across the trials, you raise \( 1-p \) to the power of the number of failures, \( n - x \).
In this example, if \( p = 0.99 \), then \( 1-p = 0.01 \). The number of failures in 40 trials with 38 successes is 2, calculated as \( 40 - 38 = 2 \). Therefore, the probability of failures is given by \[ (1-0.99)^2 = 0.01^2 \].
The result shows the probability of encountering 2 failures in these trials, which helps in computing the overall probability sought. The calculation results in \(0.01^2 = 0.0001\).
