Question

Fifteen coupons are numbered 1, 2, . . ., 15, respectively. Seven coupons are selected at random one at a time with replacement. The probability that the largest number appearing on a selected coupon is 9, is

• A. ${\left(\frac{9}{15}\right)}^7$
• B. ${\left(\frac{8}{15}\right)}^7$
• C. ${\left(\frac{9}{15}\right)}^7-{\left(\frac{8}{15}\right)}^7$
• D. None of these

Answer 1

The correct option is C ${\left(\frac{9}{15}\right)}^7-{\left(\frac{8}{15}\right)}^7$

Since there are 15 possible ways for selecting a coupon and seven coupons are selected with replacement,
The total number of ways of selecting seven coupons $={15}^7$
It is given that the maximum number on the selected coupon is 9, therefore the selection is to be made from the coupons numbered 1 to 9. This can be made in $9^7$ ways. Out of these $9^7$ cases, $8^7$ does not contain the number 9.
Thus, the favourable number of cases $=9^7-8^7$.
$\therefore$ Required probability $=\frac{9^7-8^7}{{15}^7}$