Question

43% of men consider themselves professional baseball fans. you randomly select 10 men and ask each if he considers himself a professional baseball fan. find the probability that the number who consider themselves baseball fans is (a) exactly five, (b) at least six, and (c) less than four. (a) p(5)= (round to three decimal places as needed.)

Explanation:

Step1: Identify the binomial probability formula

The binomial probability formula is \(P(x)=C(n,x)\times p^{x}\times(1 - p)^{n - x}\), where \(n\) is the number of trials, \(x\) is the number of successes, \(p\) is the probability of success on a single trial, and \(C(n,x)=\frac{n!}{x!(n - x)!}\)

Here, \(n = 10\), \(p=0.43\), and for part (a) \(x = 5\)

Step2: Calculate the combination \(C(10,5)\)

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Step3: Calculate \(P(5)\)

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\((0.43)^{5}=0.43\times0.43\times0.43\times0.43\times0.43\approx0.0147\)
\((0.57)^{5}=0.57\times0.57\times0.57\times0.57\times0.57\approx0.0525\)
\(P(5)=252\times0.0147\times0.0525\approx0.196\)

Answer:

\(0.196\)