Question

A person draws a card from a pack of playing cards, replaces it and shuffles the pack. He continues doing this until he draws a spade. The chance that he will fail exactly the first two times is?
A. \[\dfrac{9}{{64}}\]
B. \[\dfrac{1}{{64}}\]
C. \[\dfrac{1}{{16}}\]
D. \[\dfrac{9}{{16}}\]

Answer 1

Hint: Here, in questions like this, where playing cards are involved, generally we apply the same concept in all these types of questions. In these kinds of questions, we must have knowledge of cards that there are a total of 52 cards in the pack of cards. The cards deck comprises 52 cards of 4 different suits- Spades, Hearts, Clubs, and Diamonds. Each suite has 13 cards. Spades and Clubs are black-colored cards whereas Hearts and Diamonds are red-colored cards

Formula Used:
\[P = \dfrac{{{\text{Favourable number of outcomes}}}}{{{\text{Total number of outcomes}}}}\]

Complete step-by-step solution:
As already stated above, there are a total of 13 spade cards in a deck of 52 cards.
The probability of getting a single spade = Total no of possible outcomes (a spade is chosen)/ Total no of possible outcomes (any card is chosen) = \[\dfrac{{13}}{{52}} = \dfrac{1}{4}\]
Hence, the probability of not getting a spade,
\[
  P = 1 - \dfrac{1}{4} \\
   = \dfrac{3}{4} \\
 \]
Moreover, the likelihood that he will fail exactly the first two times is,
\[
  P' = \dfrac{3}{4} \times \dfrac{3}{4} \\
   = \dfrac{9}{{16}} \\
 \]

Hence, option (D) is correct

Note: In this type of question, we usually face problems while solving or using the formula. So we need to be careful while using or solving formulas. Through the proper use of formulas, solving questions becomes very simple and easy. Also, we should remain careful not to make any mistake in calculation factorial and we need to be careful while solving this. One should also keep in mind how many cards and suits are in a deck of cards as well as how many cards are in each suit because, without this knowledge, we won't be able to solve these difficulties.