What is the probability of getting two aces when two cards are drawn from a pack of 52 cards?

Choose 2 cards randomly from a deck of 52 playing cards. Let $E_1$ be the even that you picked 2 aces and $E_2$ be the event that you picked two cards with same value. Find $P(E_i)$ $i=1,2$.

ATTTEMPT:

The sample space is the set of all two-card hands. So its size is ${52 \choose 2}$. Now, lets find $|E_1|$. We only have 4 aces and we want two of them so that gives ${4 \choose 2}$ hence

$$ P(E_1) = \frac{ {4 \choose 2 }}{ {52 \choose 2 }} $$

As for the second one, notice we can let $A_k$ be the event that the two cards have kth denominations where $k$ ranges from ace to king. We can write $E_1 = \bigcup_{k=1}^{13} A_k $. Hence,

$$ P(E_1) = \sum P(A_k) $$

To find $P(A_k)$ notice first pick 2 suits we count this in ${4 \choose 2}$ ways and then kth denomination and since there are 13 of then we have

$$ P(A_k) = \frac{13 {4 \choose 2} }{ {52 \choose 2} } $$

Tehrefore,

$$ P(E_1) = \sum \frac{13 {4 \choose 2} }{ {52 \choose 2} } = \boxed{ \frac{13^2 {4 \choose 2} }{ {52 \choose 2} } } $$

Am I correct?

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

No worries! We‘ve got your back. Try BYJU‘S free classes today!