In a survey, 70% of people prefer tea over coffee. If 10 people are surveyed, wh

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Question: In a survey, 70% of people prefer tea over coffee. If 10 people are surveyed, what is the probability that exactly 7 prefer tea?

Options:

Correct Answer: 0.193

Solution:

Using binomial probability: P(X=7) = C(10,7) * (0.7)⁷ * (0.3)³ = 120 * 0.0823543 * 0.027 = 0.193.

In a survey, 70% of people prefer tea over coffee. If 10 people are surveyed, what is the probability that exactly 7 prefer tea?

In a survey, 70% of people prefer tea over coffee. If 10 people are surveyed, what is the probability that exactly 7 prefer tea?

Correct Answer: 0.193

Step 1: Understand the problem. We want to find the probability that exactly 7 out of 10 people prefer tea, given that 70% prefer tea.
Step 2: Identify the parameters. Here, n (number of trials) = 10, k (number of successes) = 7, p (probability of success) = 0.7, and q (probability of failure) = 0.3.
Step 3: Use the binomial probability formula: P(X=k) = C(n, k) * (p^k) * (q^(n-k)).
Step 4: Calculate C(10, 7), which is the number of ways to choose 7 successes from 10 trials. C(10, 7) = 10! / (7! * (10-7)!) = 120.
Step 5: Calculate (0.7)^7, which is the probability of 7 people preferring tea. (0.7)^7 = 0.0823543.
Step 6: Calculate (0.3)^3, which is the probability of 3 people preferring coffee. (0.3)^3 = 0.027.
Step 7: Combine all parts using the formula: P(X=7) = C(10, 7) * (0.7)^7 * (0.3)^3 = 120 * 0.0823543 * 0.027.
Step 8: Calculate the final probability: P(X=7) = 120 * 0.0823543 * 0.027 = 0.193.

Binomial Probability – The question tests the understanding of binomial probability distribution, specifically calculating the probability of a certain number of successes in a fixed number of 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 involves multiplying the probability of success raised to the number of successes by the probability of failure raised to the number of failures.