Question

a poll is given, showing 60% are in favor of a new building project. if 10 people are chosen at random, what is the probability that exactly 5 of them favor the new building project?

Explanation:

Step1: Identify the distribution

This is a binomial probability problem. The binomial probability formula is \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \), where \( n = 10 \) (number of trials), \( k = 5 \) (number of successes), \( p = 0.6 \) (probability of success).

Step2: Calculate the combination

First, calculate \( \binom{10}{5} \). The formula for combinations is \( \binom{n}{k} = \frac{n!}{k!(n - k)!} \). So \( \binom{10}{5} = \frac{10!}{5!5!} = \frac{10\times9\times8\times7\times6}{5\times4\times3\times2\times1} = 252 \).

Step3: Calculate the probability terms

Next, calculate \( p^k = 0.6^5 \) and \( (1 - p)^{n - k} = 0.4^{5} \). \( 0.6^5 = 0.07776 \) and \( 0.4^5 = 0.01024 \).

Step4: Multiply the terms

Multiply the combination, \( p^k \), and \( (1 - p)^{n - k} \): \( 252\times0.07776\times0.01024 \approx 0.200658 \).

Answer:

\( \approx 0.2007 \) (or 20.07% if expressed as a percentage)