Please i need an answer to the following probabilities

Question

Roll a fair die repeatedly. Let X be the number of 6’s in the first 10 rolls and let Y the number of rolls needed to obtain a 3. (a) Write down the probability mass function of X. (b) Write down the probability mass function of Y . (c) Find an expression for P(X ≥ 6). (d) Find an expression for P(Y > 10).

Answer 1

a) The probability of rolling a 6 on each roll is 1/6. There are 10 rolls. Thus, the formula for the binomial distribution is

P(X = x) = C(10,x)(1/6)^x(1-1/6)^10-x = C(10,x)(1/6)^x(5/6)^10-x or C(10,x)5^10-x/6¹⁰

Alternatively, use your calculator or Excel. In Excel, P(X = x) = binom.dist(x,10,1/6,0)

b) This is the geometric distribution.

P(Y = y) = 1/6(5/6)^y-1

c)P(X >= 6) = P(6) + P(7) + P(8) + P(9) + P(10) where we plug in the values 6 - 10 into a)

Alternatively, P(X >= 6) = 1 - P(X <= 5), and we can get the cumulative distribution from packages such as Excel. Note the 1 in the last position of the formula, indicating cumulative.

1 - binom.dist(5,10,1/6,1) = 0.002438

d) P(Y > 10) = P(non-3s on the first 10 rolls) = (5/6)¹⁰ = .1615