Answer to Question #219178 in Statistics and Probability for shoki

Question

1. At Nakumatt Supermarket, 30% of the customers pay by credit card. Find the probability that in a randomly selected sample of ten customers:

a. Exactly two pay by credit card

b. Less than three pay by credit card

c. At least four than three pay by credit card

d. More than seven pay by credit card.

Answer 1

$n=10 \\ p=0.3 \\ q=1-0.3=07 \\ P(X=x) = C^{10}_x \times 0.3^x \times 0.7^{10-x} \\ a. \; P(X=2) = C^{10}_2 \times 0.3^2 \times 0.7^{8} \\ = \frac{10!}{2!(10-2)!} \times 0.09 \times 0.05764 \\ = 0.23347 \\ b. \; P(X<3) = P(X=0)+P(X=1)+P(X=2) \\ P(X=0) = C^10_0 \times 0.3^0 \times 0.7^{10} = 0.02825 \\ P(X=1) = C^10_1 \times 0.3^1 \times 0.7^{9} = 0.12106 \\ P(X=2) = C^10_2 \times 0.3^2 \times 0.7^{8} = 0.23347 \\ P(X<3) = 0.02825 + 0.12106 + 0.23347 = 0.38278 \\ c. \; P(X≥4) = 1-P(X<4) \\ = 1 -[P(X=0)+P(X=1)+P(X=2)+P(X=3)] \\ P(X=3) = C^10_3 \times 0.3^3 \times 0.7^{7} = 0.26682 \\ = 1 -[0.02825 + 0.12106 + 0.23347 + 0.26682] \\ = 1 -0.6496 \\ =0.3504 \\ d. \;P(X>7) = P(X=8)+P(X=9)+P(X=10) \\ P(X=8) = C^10_8 \times 0.3^8 \times 0.7^{2} = 0.00144 \\ P(X=9) = C^10_9 \times 0.3^9 \times 0.7^{1} = 0.00013 \\ P(X=10) = C^{10}_{10} \times 0.3^{10} \times 0.7^{0} =0.0000059 \\ P(X>7) = 0.00144+0.00013+0.0000059 = 0.0015759$