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In a survey, 60% of people prefer tea over coffee. If 5 people are randomly sele

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Question: In a survey, 60% of people prefer tea over coffee. If 5 people are randomly selected, what is the probability that exactly 3 prefer tea?

Options:

  1. 0.2304
  2. 0.3456
  3. 0.432
  4. 0.512

Correct Answer: 0.3456

Solution: 

Using the binomial probability formula, P(X=3) = C(5,3) * (0.6^3) * (0.4^2) = 10 * 0.216 * 0.16 = 0.3456.

In a survey, 60% of people prefer tea over coffee. If 5 people are randomly sele

Practice Questions

Q1
In a survey, 60% of people prefer tea over coffee. If 5 people are randomly selected, what is the probability that exactly 3 prefer tea?
  1. 0.2304
  2. 0.3456
  3. 0.432
  4. 0.512

Questions & Step-by-Step Solutions

In a survey, 60% of people prefer tea over coffee. If 5 people are randomly selected, what is the probability that exactly 3 prefer tea?
  • Step 1: Understand the problem. We want to find the probability that exactly 3 out of 5 people prefer tea, given that 60% prefer tea.
  • Step 2: Identify the probability of success (preferring tea). This is 60%, or 0.6.
  • Step 3: Identify the probability of failure (preferring coffee). This is 40%, or 0.4.
  • Step 4: Use the binomial probability formula: P(X=k) = C(n, k) * (p^k) * (q^(n-k)), where n is the total number of trials, k is the number of successes, p is the probability of success, and q is the probability of failure.
  • Step 5: In our case, n = 5 (total people), k = 3 (people who prefer tea), p = 0.6, and q = 0.4.
  • Step 6: Calculate C(5, 3), which is the number of ways to choose 3 successes from 5 trials. C(5, 3) = 5! / (3! * (5-3)!) = 10.
  • Step 7: Calculate (0.6^3), which is the probability of exactly 3 people preferring tea. This equals 0.216.
  • Step 8: Calculate (0.4^2), which is the probability of the remaining 2 people preferring coffee. This equals 0.16.
  • Step 9: Multiply all parts together: P(X=3) = C(5, 3) * (0.6^3) * (0.4^2) = 10 * 0.216 * 0.16.
  • Step 10: Calculate the final result: 10 * 0.216 = 2.16, and then 2.16 * 0.16 = 0.3456.
  • Binomial Probability – The question tests the understanding of the binomial probability formula, which calculates the likelihood of a specific number of successes in a fixed number of independent Bernoulli trials.
  • Combinations – The use of combinations (C(n, k)) to determine the number of ways to choose k successes from n trials is a key concept in solving the problem.
  • Probability Calculation – The calculation of probabilities for success (tea preference) and failure (coffee preference) is essential for applying the binomial formula.
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