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Chapter 10: Problem 102

Sixty-five percent of all male voters and \(40 \%\) of all female voters favor a particular candidate. A sample of 100 male voters and another sample of 100 female voters will be polled. What is the probability that at least 10 more male voters than female voters will favor this candidate?

Short Answer

Expert verified
The probability that at least 10 more male voters than female voters will favor the candidate is computed by summing probabilities using binomial distribution formula from \(k=60\) to \(k=100\).

Step by step solution

01

Calculate Probability of Male Voter Supporting Candidate

Given that 65% of all male voters favor the candidate, the probability of a male voter supporting the candidate is \(0.65\). This is denoted as \(p_m = 0.65\) where \(p_m\) is the probability.
02

Calculate Probability of Female Voter Supporting Candidate

Similarly, since 40% of all female voters favor the candidate, the probability of a female voter supporting the candidate is \(0.40\). This is denoted as \(p_f= 0.40\) where \(p_f\) is the probability.
03

Calculate the Probability of Interest Using Binomial Distribution

The problem asks for the probability of at least 10 more male voters than female voters favoring the candidate. This translates to finding the probability of having 10 more successes in the male sample than in the female sample. Here success is defined as a vote in favor of the candidate. This problem can be solved using the binomial distribution formula: \(P(k; n, p) = C(n, k) * p^k * (1-p)^{n-k}\) where \(C(n, k)\) is the combination of \(n\) items taken \(k\) at a time. In our case, \(n\) is 100 and \(k\) should range from a minimum of 60 (as 50 females would vote in favor at most) to 100. We calculate the desired probability by summing the probabilities of these outcomes.
04

Use Binomial Distribution and Summation

Compute the sum from \(k=60\) to \(k=100\) of the probabilities calculated using the binomial distribution formula. This gives the probability that at least 10 more male voters than female voters will favor the candidate.

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