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 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 replacing the card. So our sample space will not change

In a deck of 52 cards, there are 13 hearts.

Let X be the random variable denoting the number of hearts drawn for an event when 3 cards are drawn successively with replacement.

∴ X can take values 0 , 1, 2 or 3

P(X = 0) =

{As we have to select 1 card at a time such that no heart is there so the first probability is 39/52 and as the drawn card is replaced, next time again probability is 39/52 and again the same thing }

Similarly, we proceed for other cases.

P(X = 1) =

{we might get a heart in the first card or second card or third, so probabilities in all 3 cases are added as they are mutually exclusive events}

Similarly,

P(X = 2) =

{First 2 cards are heart and 3^rd is non-heart, last 2 are hearts and so on cases. As these cases are mutually exclusive. Hence they are added }

Similarly,

P(X = 3) =

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: