Final answer:
To find the probability of P=(Y=2X+a) where a=3, we need to understand the probability distribution of X and Y. X represents the number of rolls needed to see the first 6, and Y represents the number of rolls needed to see the second 6. Using the geometric distribution, we can calculate the probability of P=(Y=2X+a) = P(Y=5) = (5/6)^4 * (1/6).
Step-by-step explanation:
To find the probability of P=(Y=2X+a) where a=3, we need to understand the probability distribution of X and Y.
Since X represents the number of rolls needed to see the first 6, it follows a geometric distribution with parameter p=1/6 (since there is a 1/6 probability of rolling a 6 on each roll).
Similarly, Y represents the number of rolls needed to see the second 6, counting from the first roll. Since the first 6 already occurred on the first roll (X=1), we can think of Y as the number of additional rolls needed to see the second 6, which also follows a geometric distribution with parameter p=1/6.
Now, let's calculate the probability of P=(Y=2X+a), where a=3:
First, let's find the probability P(X=1). Since X follows a geometric distribution, P(X=1) = p = 1/6.
Next, let's find the probability P(Y=2X+a) = P(Y=2(1)+3) = P(Y=5).
Since Y follows a geometric distribution with parameter p=1/6, we can calculate P(Y=5) = (1-p)^4 * p = (5/6)^4 * (1/6).
Therefore, the probability of P=(Y=2X+a) = P(Y=5) = (5/6)^4 * (1/6).