Question

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.

Answer 1

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)= $nC_{x}p^{x}q^{n-x}$

a)P(10)= $nC_{10}$ $p^{10}q^{n-10}$ = $18C_{10}*0.26^{10} *0.74^{18-10} ≈ 0.00555$

P(18)= $nC_{18}$ $p^{18}q^{n-18}$ =$18C_{18}*$ $0.26^{18} *0.74^{18-18}$ ≈ 0.0000000000295.

P(3)= $nC_{3}$ $p^{3}q^{n-3}$ = $18C_{3}*$ $0.26^{3}* 0.74^{18-3}$ ≈ 0.157

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

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

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