Textbook Question
"Problems 17–28: In each problem, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in n independent trials of the experiment.
n = 11, p = 0.75, x ≥ 8"
5. Binomial Distribution & Discrete Random Variables
Binomial Distribution
2:05 minutesProblem 6.T.7a
Textbook Question
"According to a study conducted by CESI Debt Solutions, 80% of married people hide purchases from their mates. In a random sample of 20 married people, find and interpret:
a. The probability exactly 15 hide purchases from their mates.
"
Verified step by step guidance1
Identify the type of probability distribution involved. Since we are dealing with a fixed number of trials (20 married people), each with two possible outcomes (hide purchase or not), and a constant probability of success (80%), this is a binomial distribution problem.
Define the parameters of the binomial distribution: the number of trials \(n = 20\), the probability of success on each trial \(p = 0.8\), and the number of successes we want to find the probability for \(k = 15\).
Recall the binomial probability formula:
\[P(X = k) = \binom{n}{k} p^{k} (1-p)^{n-k}\]
where \(\binom{n}{k}\) is the binomial coefficient representing the number of ways to choose \(k\) successes out of \(n\) trials.
Calculate the binomial coefficient \(\binom{20}{15}\), which counts the number of ways to select exactly 15 people who hide purchases out of 20.
Substitute the values into the formula:
\[P(X = 15) = \binom{20}{15} (0.8)^{15} (0.2)^{5}\]
This expression gives the probability that exactly 15 out of 20 married people hide purchases from their mates.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success. Here, 'success' is a married person hiding purchases, with probability 0.8, and the number of trials is 20. It helps calculate the probability of exactly k successes.
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Mean & Standard Deviation of Binomial Distribution
Probability Mass Function (PMF)
The PMF of a binomial distribution gives the probability of observing exactly k successes in n trials. It is calculated using the formula: P(X = k) = C(n, k) * p^k * (1-p)^(n-k), where C(n, k) is the combination of n items taken k at a time.
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Interpretation of Probability
Interpreting the probability means explaining what the calculated value represents in context. For example, the probability of exactly 15 people hiding purchases indicates how likely it is to observe that specific outcome in a random sample of 20 married people.
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