For a particular clothing store, a marketing firm finds that 28% of $10-off coupons delivered by mail are redeemed. Suppose that 16 customers are randomly selected and are mailed $10-off coupons. Round your solutions for this exhibit to 4 decimal places.

1)What is the probability that five of the customers redeem the coupon?

2)What is the probability that more than 8 customers redeem the coupon?

3)What is the variance of the number of coupons that will be redeemed?

(a) 0.2026

(b) 0.0163

(c) 3.2256

Step-by-step explanation

Given that probability p = 0.28 

Number of customers (n) = 16

We have to use binomial formula P(X=r)= ((n−r)!∗r!n!?)∗pr∗(1−p)n−r

(a) We need to find P(X=5)

P(X=5)= ((16−5)!∗5!16!?)∗0.285∗(1−0.28)16−5 =4368∗0.001721037∗0.026956125 =0.2026

(b) We need to find P(X>8)

P(X>8)=P(X=9)+P(X=10)+P(X=11)+P(X=12)+P(X=13)+P(X=14)+P(X=15)+P(X=16) =((16−9)!∗9!16!?)∗0.289∗(1−0.28)16−9 + ((16−10)!∗10!16!?)∗0.2810∗(1−0.28)16−10 + ((16−11)!∗11!16!?)∗0.2811∗(1−0.28)16−11 + ((16−12)!∗12!16!?)∗0.2812∗(1−0.28)16−12 + ((16−5)!∗5!16!?)∗0.2813∗(1−0.28)16−13 + ((16−14)!∗14!16!?)∗0.2814∗(1−0.28)16−14 + ((16−15)!∗15!16!?)∗0.2815∗(1−0.28)16−15 + ((16−16)!∗16!16!?)∗0.2816∗(1−0.28)16−16 =(11440∗0.000010578456∗0.10030613) + (8008∗0.000002961968∗0.13931407) + (4368∗0.000000829351∗0.193491763) + (1820∗0.000000232218∗0.26873856) + (560∗0.000000065021∗0.373248) + (120∗0.000000018206∗0.5184) + (16∗0.000000005098∗0.72) + (1∗0.000000001427∗1) =0.01214+0.0033+0.0007+0.00011+0.00001+0.0000+0.0000+0.0000 =0.0163

(c) Variance = n*p*(1-p)

= 16*0.28*(1-0.28)

= 16*0.28*0.72

= 3.2256