Answer to Question #127612 in Statistics and Probability for Harsha

Question #127612
Supermarket CRM system has past records;
a supermarket finds that 26% of people who enter the supermarket will make a purchase. 18 people enter the supermarket during a one-hour period.
(a) What is the probability that exactly 10 customers, 18 customers and 3 customers make a purchase?
(b) Find the expected number of customers who make a purchase.
(c) Find the variance and standard deviation of the number of customers who make a purchase.
1
Expert's answer
2020-08-03T19:16:26-0400

Since the experiment consists of n identical trials.Each trial results in one of the two outcomes, called success and failure.The probability of success, denoted p, remains the same from trial to trial.The n trials are independent ,we use binomial distribution.

Formula: P(x)= nCxpxqnxnC_{x}p^{x}q^{n-x}


a)P(10)= nC10nC_{10} p10qn10p^{10}q^{n-10} = 18C100.26100.7418100.0055518C_{10}*0.26^{10} *0.74^{18-10} ≈ 0.00555


P(18)= nC18nC_{18} p18qn18p^{18}q^{n-18} =18C1818C_{18}* 0.26180.7418180.26^{18} *0.74^{18-18} ≈ 0.0000000000295.


P(3)= nC3nC_{3} p3qn3p^{3}q^{n-3} = 18C318C_{3}* 0.2630.741830.26^{3}* 0.74^{18-3} ≈ 0.157


b)Expected value µ = np = 18 * 0.26 = 4.68


c) Variance:σ2=npq=180.260.74=3.46Variance: σ^ 2 = npq = 18 *0.26 * 0.74 = 3.46

StandardDeviation:σ=npq=3.46=1.86Standard Deviation: σ =\sqrt{npq}= \sqrt{3.46} = 1.86



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