Question

From a pack of 52 cards, two cards are drawn at random one after the other with replacement. What is the probability that both cards are kings?
A. \[\dfrac{1}{{169}}\]
B. \[\dfrac{2}{{169}}\]
C. \[\dfrac{5}{{169}}\]
D. None of these

Answer 1

Hint: First of all, find the probability of getting a king for the first card and then replace that card. Then find the probability of getting a king for the second card and use the multiplication rule of probability to find the required answer.

Complete step-by-step answer:

Total number of cards = 52
Total number of kings = 4
Given, two cards are drawn at random one after the other with replacement.
We know that the probability of an event \[E\] is given by \[P\left( E \right) = \dfrac{{{\text{Number of favorable outcomes}}}}{{{\text{Total number of outcomes}}}}\]
The number of favorable outcomes for first card = 4
The total number of outcomes = 52
Thus, the probability of getting kings in the first card \[ = \dfrac{4}{{52}} = \dfrac{1}{{13}}\]
Now, the card is replaced.
The number of favorable outcomes for second card = 4
The total number of outcomes = 52
Thus, the probability of getting kings in the second card \[ = \dfrac{4}{{52}} = \dfrac{1}{{13}}\]
By using the multiplication rule of probability, the probability that both cards are kings \[ = \dfrac{1}{{13}} \times \dfrac{1}{{13}} = \dfrac{1}{{169}}\]
Thus, the probability that both cards are kings is A. \[\dfrac{1}{{169}}\]

Note: The probability of an event \[E\] is always greater than or equal to zero and less than or equal to one i.e., \[0 \leqslant P\left( E \right) \leqslant 1\]. Rule of multiplication of probability that the events A and B both occurs equal to the probability that event A occurs times the probability that B occurs, given that A has occurred.