Question

Two cards are drawn successively with replacement from a well shuffled deck of $52$ cards. Find the probability distribution of the number of aces.
A). $\dfrac{1}{{169}}$
B). $\dfrac{1}{{221}}$
C). $\dfrac{1}{{265}}$
D). $\dfrac{4}{{663}}$

Answer 1

In the given question, we have been given to find the probability distribution of a card from a deck of card. It has been given that there is replacement in the cards, so we have to consider taking the whole deck in all the trials. So, we just have to apply the formula of probability for the two cases, apply the required arithmetic operator, and find the answer.

Formula used:
We are going to use the formula of probability:
Consider we have to find the probability of an event $E$, given that the number of favorable events are $F\left( E \right)$ and the total number of events are $T\left( E \right)$,
$P\left( E \right) = \dfrac{{F\left( E \right)}}{{T\left( E \right)}}$

Complete step by step solution:
In the given question, we had to find the probability distribution of the number of aces.
Here, the draws are made from the deck after replacement, i.e., after each draw, the whole new deck is taken.
Now, the probability of drawing an ace from a deck,
Number of aces in a deck, $F\left( E \right) = 4$
Total number of cards in a deck, $T\left( E \right) = 52$
So, the probability is given by,
$P\left( {{E_1}} \right) = \dfrac{4}{{52}} = \dfrac{1}{{13}}$
Again, when the card is drawn, the same probability is there, i.e., $P\left( {{E_2}} \right) = \dfrac{1}{{13}}$.
So, the total probability is,
$P\left( E \right) = \dfrac{1}{{13}} \times \dfrac{1}{{13}} = \dfrac{1}{{169}}$

Note: In the given question, we had to find the probability of a particular card which was drawn from a deck of card and after each draw, there was replacement – after each draw, the result was noted, and the card was put back into the deck. To solve that, we just used the formula of probability for the given cases, applied the required arithmetic operator (multiplication) and found the answer.