Question

Fifteen coupons are numbered 1 to 15. Seven coupons are selected at random, one at a time with replacement. The probability that the largest number appearing on a selected coupon be 9 is
A. $ {\left( {\dfrac{1}{{15}}} \right)^7} $
B. $ {\left( {\dfrac{8}{{15}}} \right)^7} $
C. $ {\left( {\dfrac{3}{5}} \right)^7} $
D.None of these

Answer 1

Hint: It is known that when $ r $ objects out of $ n $ are selected at random with replacement, the total number of ways to do so is given by $ {n^r} $ .
Also, to find the probability of an event, use the formula $ {\rm{Probability = }}\dfrac{{{\rm{Favorable cases}}}}{{{\rm{Total number of cases}}}} $

Complete step-by-step answer:
We know that, when $ r $ objects out of $ n $ are selected at random with replacement, the total number of ways to do so is given by $ {n^r} $ .
On applying the same concept, we have that 7 coupons are selected from 15 at random with replacement, so $ n = 15 $ and $ r = 7 $
Hence, the total number of ways to select 7 coupons at random with replacement is given by $ {15^7} $ .
For 9 to be the largest amongst the selected coupons, the coupons must be selected from 1, 2, 3, ….,9.
From these 9 coupons only 7 are to be selected with replacement. So, in this case, $ n = 9 $ and $ r = 7 $ .
This can be done in $ {9^7} $ ways.
But there may be cases in which the coupon bearing number 9 won’t be selected.
In such a case, the highest number would be 8.
So, the number of ways to select 7 coupons from 8 coupons with replacement is given by $ {8^7} $ .
But the required cases should contain only the cases where the largest number on the coupon should be 9. So, we shall have to subtract the cases where 9 is not the largest from the cases where 9 is the largest.
The number of cases in which the 9 is largest is given by $ {9^7} - {8^7} $ .
In this question, we are asked to find the probability that the largest number appearing on the coupon is 9.
We have to apply the formula i.e.
$\Rightarrow {\rm{Probability = }}\dfrac{{{\text{Favorable cases}}}}{{{\rm{Total number of cases}}}} $
The probability that the largest number appearing on the coupon is 9 $ {\rm{ = }}\dfrac{{{9^7} - {8^7}}}{{15{}^7}} $ .

So, the correct answer is “Option D”.

Note: Students should take care of the language used in this question. There is a possibility that students might misunderstand the language while calculating the value of probability.
When $ r $ objects out of $ n $ are selected at random with replacement, the total number of ways to do so is given by $ {n^r} $ . Students should take care while using this formula, they often mistake the value to be $ {r^n} $ instead of $ {n^r} $ .