Question

Roll a fair die repeatedly. Let X denote the number of 4’s in the first 6 rolls and let Y denote

the number of rolls needed to obtain a 2.
a. Write down the probability mass function of X.
b. Write down the probability mass function of Y.
c. Find an expression for P( ≥ 3).
d. Find an expression for P( > 6).

Answer 1

To answer the student's question, the PMF of X is found using the binomial distribution, and the PMF of Y is found using the geometric distribution. P(Y ≥ 3) and P(Y > 6) can be calculated using geometric series and the complement rule.

Step-by-step explanation:

Rolling a fair six-sided die involves computing probabilities for various outcomes. Here's how to approach the student's question:

1. The probability mass function (PMF) of X, which is the number of 4's rolled in the first 6 rolls, is a binomial distribution as each roll is independent. Therefore, PMF of X is P(X=k) = (6 choose k)*(1/6)^k*(5/6)^(6-k), where k can be from 0 to 6.
2. For Y, the number of rolls to get a 2, the PMF of Y follows a geometric distribution. Thus, P(Y=k) = (1/6)*(5/6)^(k-1), where k is the k-th roll.
3. To find P(Y ≥ 3), we calculate 1 - P(Y=1) - P(Y=2), which accounts for it taking at least three rolls to get a 2.
4. To find P(Y > 6), we sum the geometric series starting from the 7th term to infinity, or use the formula for the complement, which is 1 - (sum of probabilities for k = 1 to 6).

These calculations use concepts from binomial and geometric distributions and provide a step-wise approach to solving these probability questions.