A retail store surveys customers and determines that the probability of any customer making a purchase is 0.3. For a random sample of 10 customers:
(a) Identify the parameters \(n\) and \(p\) for the binomial model.
(b) Calculate the probability that exactly 4 customers make a purchase.
(c) Determine the probability that at least 3 customers make a purchase.
Question
Identify the parameters \(n\) and \(p\) used in the binomial distribution for the given scenario.
Hint
Hint
Hint
Determine the number of trials (customers) and the probability of success (making a purchase).
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Complete Answer
In this case, \(n = 10\) and \(p = 0.3\).
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Question
Calculate the probability that exactly 4 customers make a purchase using the binomial probability formula.
Hint
Hint
Hint
Apply the formula \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\) with \(k=4\).
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Complete Answer
The probability is \(\binom{10}{4}(0.3)^4(0.7)^6\), which evaluates to approximately 0.2001.
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Question
Determine the probability that at least 3 customers make a purchase.
Hint
Hint
Hint
Express \(P(X \ge 3)\) as \(1 – P(X \le 2)\) and use the binomial formula or calculator’s cumulative binomial function.
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Complete Answer
The probability is \(1 – \left[\binom{10}{0}(0.3)^0(0.7)^{10} + \binom{10}{1}(0.3)^1(0.7)^9 + \binom{10}{2}(0.3)^2(0.7)^8\right]\), which is approximately 0.8497.
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