Question

a large fast - food restaurant is having a promotional game where game pieces can be found on various products. customers can win food or cash prizes. according to the company, the probability of winning a prize (large or small) with any eligible purchase is 0.15. consider your next 26 purchases that produce a game piece. calculate the following: this is a binomial distribution. round your answers to at least 4 decimal places. a) what is the probability that you win 4 prizes? b) what is the probability that you win more than 6 prizes? c) what is the probability that you win 2 prizes or fewer?

Explanation:

Step1: Identify the binomial formula

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

For this problem, \(n = 26\), \(p=0.15\), \(1-p = 0.85\)

Part a)

Step1: Calculate \(C(26,4)\)

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Step2: Calculate \(P(X = 4)\)

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Part b)

Step1: Use the complement rule

\(P(X>6)=1 - P(X\leq6)\)

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Step2: Calculate \(P(X>6)\)

\(P(X>6)=1 - 0.8443=0.1557\)

Part c)

Step1: Calculate \(P(X\leq2)\)

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$$ P(X\leq2)=0.0100 + 0.0430+0.0998=0.1528 $$

Answer:

a) \(0.2174\)
b) \(0.1557\)
c) \(0.1528\)