MCQ

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 is 9 is

#### Options

\[\left( \frac{3}{5} \right)^7 \]

\[\left( \frac{1}{15} \right)^7\]

\[\left( \frac{8}{15} \right)^7\]

None of these

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#### Solution

Answer: None Of these

Let p= probability that a selected coupon bears number \[\leq 9\] .

\[p = \frac{9}{15} = \frac{3}{5}\]

n = number of coupons drawn with replacement

X = number of coupons bearing number \[\leq 9\]\ Probability that the largest number on the selected coupons does not exceed 9

= probability that all the coupons bear number \[\leq 9\]

X = number of coupons bearing number \[\leq 9\]\ Probability that the largest number on the selected coupons does not exceed 9

= probability that all the coupons bear number \[\leq 9\]

= P(X=7) = \[^ {7}{}{C}_7 p^7 q^0 = \left( \frac{3}{7} \right)^7\]

Similarly, probability that largest number on the selected coupon bears the number \[\leq 8\] will be

P(X=7) = \[^{7}{}{C}_7 p^7 q^0 = \left( \frac{8}{15} \right)^7\]

P(X=7) = \[^{7}{}{C}_7 p^7 q^0 = \left( \frac{8}{15} \right)^7\]

(since, p will become \[\frac{8}{15}\])

Hence required probability will be =

\[\left( \frac{3}{7} \right)^7 - \left( \frac{8}{15} \right)^7\]

Concept: Bernoulli Trials and Binomial Distribution

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