Let 𝑋X and 𝑌Y be independent and identically distributed Geometric random variables with parameter 0.80.8.Find 𝑃(𝑋=4∣𝑋+𝑌=8).P(X=4∣X+Y=8).Enter the answer correct to two decimal places.

Given that X and Y are independent and identically distributed Geometric random variables with parameter 0.8, we want to find P(X=4 | X+Y=8).

Step 1: Identify the probability mass function (pmf) of a Geometric distribution. The pmf of a Geometric distribution is given by P(X=k) = p(1-p)^(k-1), where p is the success probability and k is the number of trials needed to get the first success.

Step 2: Substitute the given values into the pmf. Here, p=0.8 and k=4, so P(X=4) = 0.8(1-0.8)^(4-1) = 0.8(0.2)^3 = 0.0064.

Step 3: Find P(X+Y=8). Since X and Y are independent, the sum of two Geometric random variables follows a Negative Binomial distribution. The pmf of a Negative Binomial distribution is given by P(X+Y=k) = C(k-1, r-1)p^r(1-p)^(k-r), where r is the number of successes (which is 2 here because we have two Geometric random variables), p is the success probability, and k is the total number of trials. Here, p=0.8, r=2, and k=8, so P(X+Y=8) = C(8-1, 2-1)(0.8)^2(1-0.8)^(8-2) = 7(0.64)(0.2)^6 = 0.007168.

Step 4: Find P(X=4 | X+Y=8) using the definition of conditional probability, which is P(A | B) = P(A ∩ B) / P(B). Here, A is the event that X=4 and B is the event that X+Y=8. Since X and Y are independent, P(A ∩ B) = P(A)P(B). So, P(X=4 | X+Y=8) = P(X=4)P(X+Y=8) / P(X+Y=8) = 0.0064 / 0.007168 = 0.8929.

So, P(X=4 | X+Y=8) = 0.89 (rounded to two decimal places).