Question

15 coupons are numbered 1, 2, 3, 4,.....15 respectively. 7 coupons are selected at random one at a time with replacement. The probability that the largest number appearing on the selected coupon is 9 is
(a) ${{\left( \dfrac{9}{16} \right)}^{6}}$
(b) ${{\left( \dfrac{8}{15} \right)}^{7}}$
(c) ${{\left( \dfrac{3}{5} \right)}^{7}}$
(d) $\dfrac{{{9}^{7}}-{{8}^{7}}}{{{15}^{7}}}$

Answer 1

Hint: First, we are given with the direct condition that 7 coupons are to be selected from 15 coupons with replacement making the condition that for every coupon selection, we have 15 ways. Then, we are given the condition that 9 is to be the largest from the selected ones which gives the range of selection from 1 to 9 only. Then, the above ways for selecting 7 coupons from 9 also selects the condition where 9 is not selected at all and thus the highest number selected in that condition becomes 8. Then, for favourable cases, we need to subtract the number of ways of getting 8 as highest number from the number of cases giving 9 as highest number, we get the final answer.

Complete step-by-step answer:
In this question, we are supposed to find the probability that the largest number appearing on the selected coupon is 9 when 7 coupons are selected at random one at a time with replacement from 15 coupons are numbered 1, 2, 3, 4,.....15 respectively.
Now, we are given the direct condition that 7 coupons are to be selected from 15 coupons with replacement making the condition that for every coupon selection, we have 15 ways.
So, for selecting 7 coupons with replacement from 15 coupons, the total ways we have are:
${{15}^{7}}$
Now, we are given the condition that 9 is to be the largest from the selected ones which gives the range of selection from 1 to 9 only.
Similarly, if the range is 1 to 9 and we have the chance to select 7 coupons from 9 total coupons with replacement gives the total number of cases as:
${{9}^{7}}$
However, the above ways for selecting 7 coupons from 9 also selects the condition where 9 is not selected at all and thus the highest number selected in that condition becomes 8.
So, we got a new condition in which 8 coupons are selected in 7 chances with replacements gives the total number of cases as:
${{8}^{7}}$
Now, we are mentioned in the question that we need only those cases which give 9 as the largest number.
So, for favourable cases, we need to subtract the number of ways of getting 8 as highest number from the number of cases giving 9 as highest number as:
${{9}^{7}}-{{8}^{7}}$
Then, the probability of selecting 9 as the highest number from the total of 15 coupons with selecting chances as 7 with replacement is given by:
$\dfrac{{{9}^{7}}-{{8}^{7}}}{{{15}^{7}}}$
Hence, option (d) is correct.

Note: Now, to solve these types of questions we need to know some basics of the probability so that we can easily solve these types of questions. So, probability is basically the ratio of the favourable cases to the total number of cases which is used in this question appropriately.