8 cards are drawn from a standard deck of cards with replacement let X=no of diamonds observed find P(X=3)

To find the probability of drawing exactly 3 diamonds when 8 cards are drawn from a standard deck of cards with replacement, we need to calculate the probability of getting a diamond (success) in each draw.

Let's break down the problem step by step.

Step 1: Determine the probability of drawing a diamond from a standard deck of cards.
- A standard deck of cards contains 52 cards, and 13 of them are diamonds.
- Therefore, the probability of drawing a diamond from the deck is 13/52, which simplifies to 1/4.

Step 2: Determine the probability of not drawing a diamond from a standard deck of cards.
- The probability of not drawing a diamond from the deck is the complement of drawing a diamond, which is 1 - 1/4 = 3/4.

Step 3: Calculate the probability of drawing exactly 3 diamonds in 8 draws.
- Since the draws are made with replacement, each draw is independent, and the probability of drawing a diamond remains constant in each draw.
- We need exactly 3 diamonds out of the 8 draws, so we multiply the probability of drawing a diamond (1/4) by itself 3 times, and multiply the probability of not drawing a diamond (3/4) by itself 5 times (since we need 5 non-diamonds as well, given 8 total draws).
- The total probability of drawing exactly 3 diamonds is: (1/4)^3 * (3/4)^5 = (1/64) * (243/1024) = 243/65536.

So, the probability (P(X=3)) of observing exactly 3 diamonds when 8 cards are drawn with replacement from a standard deck of cards is 243/65536.