Question

Seven coupons are selected at random one at a time with replacement from 15 coupons numbered 1 to 15. The probability that the largest number appearing on a selected coupon is 9, is

• A. ${\left(\frac{9}{16}\right)}^6$
• B. ${\left(\frac{8}{15}\right)}^7$
• C. ${\left(\frac{3}{5}\right)}^7$
• D. ${\left(\frac{3}{5}\right)}^7$ - ${\left(\frac{8}{15}\right)}^7$

Answer 1

The correct option is D ${\left(\frac{3}{5}\right)}^7$ - ${\left(\frac{8}{15}\right)}^7$

Each coupon can be selected in 15 ways. The total number of ways of choosing 7 copouns is ${15}^7$. If largest number is 9, then the selected numbers have to be from 1 to 9 excluding those consisting of only 1 to 8.
Probability desired is $\frac{9^7-8^7}{{15}^7}$
$={\left(\frac{3}{5}\right)}^7-{\left(\frac{8}{15}\right)}^7$