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Chapter 6: Problem 25

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) success in the \(n\) independent trials of the experiment. \(n=7, p=0.5, x>3\)

Short Answer

Expert verified
The probability of more than 3 successes is 0.609375.

Step by step solution

01

Identify the Binomial Probability Formula

The binomial probability formula is given by \[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \] where \[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \] represents the number of combinations.
02

Determine the Cumulative Probability

In this case, we need the probability of having more than 3 successes, which means we need to calculate the summation of probabilities for x = 4, 5, 6, and 7. This can be written as: \[ P(X > 3) = P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7) \]
03

Compute Individual Probabilities

Compute each individual probability using the binomial formula: For \( x = 4 \):\[ P(X = 4) = \binom{7}{4} (0.5)^4 (0.5)^{7-4} \]For \( x = 5 \):\[ P(X = 5) = \binom{7}{5} (0.5)^5 (0.5)^{7-5} \]For \( x = 6 \):\[ P(X = 6) = \binom{7}{6} (0.5)^6 (0.5)^{7-6} \]For \( x = 7 \):\[ P(X = 7) = \binom{7}{7} (0.5)^7 (0.5)^{7-7} \]
04

Calculate Each Combination

Compute the combinations: \[ \binom{7}{4} = 35 \]\[ \binom{7}{5} = 21 \]\[ \binom{7}{6} = 7 \]\[ \binom{7}{7} = 1 \]
05

Substitute Values and Calculate

Substitute the values into the formula and calculate each probability:\[ P(X = 4) = 35 \times (0.5)^4 \times (0.5)^3 = 35 \times 0.0078125 = 0.2734375 \]\[ P(X = 5) = 21 \times (0.5)^5 \times (0.5)^2 = 21 \times 0.03125 = 0.21875 \]\[ P(X = 6) = 7 \times (0.5)^6 \times (0.5)^1 = 7 \times 0.015625 = 0.109375 \]\[ P(X = 7) = 1 \times (0.5)^7 \times (0.5)^0 = 1 \times 0.0078125 = 0.0078125 \]
06

Sum the Probabilities

Add all the individual probabilities together to get the total probability of having more than 3 successes:\[ P(X > 3) = 0.2734375 + 0.21875 + 0.109375 + 0.0078125 = 0.609375 \]

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Key Concepts

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

Binomial Distribution
The binomial distribution is a discrete probability distribution. It describes the number of successes in a fixed number of independent trials. Each trial must have only two possible outcomes: success or failure.

To fully define a binomial distribution, you need two parameters:

  • The number of trials (\( n \)
  • The probability of success in a single trial (\( p \)
In our example, with \( n = 7 \) and \( p = 0.5 \), we want to find the probability of getting more than 3 successes out of 7 trials. This means we are interested in \( P(X > 3) \).
Probability Formula
The binomial probability formula is essential for solving any problem involving a binomial distribution. It allows us to calculate the probability of exactly \( x \) successes in \( n \) trials:

\[ P(X = x) = \binom{n}{x} p^x (1-p)^{n-x} \]

In the formula:

  • \( \binom{n}{x} \) is the number of ways to choose \( x \) successes out of \( n \)
  • \( p \) is the probability of success in each trial
  • \( 1-p \) is the probability of failure in each trial
To find \( P(X > 3) \), we need to compute the cumulative probability by summing up the individual probabilities for all successful outcomes greater than 3 (i.e., \( x = 4, 5, 6, \) and \( 7 \)).
Combinations
The concept of combinations is crucial in binomial probability calculations. A combination represents the number of ways to choose a subset of items from a larger set, without considering the order. In mathematical terms, it is denoted by \( \binom{n}{x} \) and calculated using the formula:

\[ \binom{n}{x} = \frac{n!}{x!(n-x)!} \]

Here, \( n! \) represents the factorial of \( n \) (the product of all positive integers up to \( n \)), and similarly for \( x! \) and \( (n-x)! \).

In our problem:

  • \( \binom{7}{4} = 35 \)
  • \( \binom{7}{5} = 21 \)
  • \( \binom{7}{6} = 7 \)
  • \( \binom{7}{7} = 1 \)
These values are essential to calculate the individual probabilities using the binomial probability formula. Once you have these combinations, you substitute them into the formula and sum the results to find the total probability.

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Most popular questions from this chapter

Determine whether the distribution is a discrete probability distribution. If not, state why. $$\begin{array}{|l|l|} \hline x & f(x) \\ \hline 10 & 0.1 \\ \hline 20 & 0.23 \\ \hline 30 & 0.22 \\ \hline 40 & 0.6 \\ \hline 50 & -0.15 \end{array}$$

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) success in the \(n\) independent trials of the experiment. \(n=40, p=0.99, x=38\)

A binomial probability experiment is conducted with the given parameters. Compute the probability of \(x\) success in the \(n\) independent trials of the experiment. \(n=9, p=0.2, x \leq 3\)

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