SATHEE: Probability 3 Question 42

Probability 3 Question 42

42. Cards are drawn one by one at random from a well shuffled full pack of 52 playing cards until 2 aces are obtained for the first time. If $N$ is the number of cards required to be drawn, then show that

$P_{r} N = n = \frac{(n - 1) (52 - n) (51 - n)}{50 \times 49 \times 17 \times 13}$

where, $2 < n \leq 50$ .

(1983, 3M)

Show Answer

Answer:

Correct Answer: 42. 0.6976

Solution:

$P (N$ th draw gives 2 nd ace $)$

$= P 1 $ a c e a n d $ (n - 2) $ o t h e r c a r d s a r e d r a w n i n $ (N - 1) $ d r a w s $ \times P N $ t h d r a w i s 2 n d a c e $$

$\begin{aligned} = \frac{4 \cdot (48)! \cdot (n - 1)! (52 - n)!}{(52)! \cdot (n - 2)! (50 - n)!} \cdot \frac{3}{(53 - n)} \\ = \frac{4 (n - 1) (52 - n) (51 - n) \cdot 3}{52 \cdot 51 \cdot 50 \cdot 49} \\ = \frac{(n - 1) (52 - n) (51 - n)}{50 \cdot 49 \cdot 17 \cdot 13} \end{aligned}$

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Probability 3 Question 42

42. Cards are drawn one by one at random from a well shuffled full pack of 52 playing cards until 2 aces are obtained for the first time. If N is the number of cards required to be drawn, then show that

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42. Cards are drawn one by one at random from a well shuffled full pack of 52 playing cards until 2 aces are obtained for the first time. If $N$ is the number of cards required to be drawn, then show that