Note: While reading this question you might have observed that cards are being drawn successively and sometimes in other question you might have get cards are drawn simultaneously

You have to be careful regarding these two words while solving the question. Both have a different meaning.

E.g.:-

When you draw 3 cards out of 52 simultaneously, then total no of ways of drawing is simply 52C₃ = 22100. You draw 3 cards at a time.

When you draw 3 cards out of 52 successively, then total no of ways of drawing is not simply 52C₃, once you have drawn the first card, only 51 cards are remaining and so on. You are drawing only one card at a time.

In such case total ways of drawing 3 cards = 52C₁ × 51C₁ × 50C_{1 =} 132600

Hence both are entirely different. So Be cautious regarding it.

Let’s solve the problem now:

Note: In our problem, it is given that after every draw, we are not replacing the card. So our sample space will change in this case

In a deck of 52 cards, there are 4 aces each of one suit respectively.

Let X be the random variable denoting the number of aces for an event when 2 cards are drawn successively.

∴ X can take values 0, 1 or 2

P(X = 0) =

{As we have to select 1 card at a time such that no ace is there so first probability is 48/52 and as the drawn card is not replaced, next time probability is 47/51 because now one card is not in our sample space and that card is not from group of ace, So now ace cards are 47 }

Similarly, we proceed for other cases.

P(X = 1) =

{we might get ace in the first card or in the second card, so both probabilities are added}

Similarly,

P(X = 2) =

Now we have p_i and x_i.

∴ Now we are ready to write the probability distribution for X:-

The following table gives probability distribution: