Question

A fair 4-sided die is rolled, let X denote the outcome. After that, if X = x, then x fair coins are tossed, let Y denote the number of Tails observed. a) Find P( X >= 3 | Y = 0 ). b) Find E( X | Y = 2 ). “Hint”: Construct the joint probability distribution for ( X, Y ) first. Write it in the form of a rectangular array with x = 1, 2, 3, 4 and y = 0, 1, 2, 3, 4.

Answer 1

a) The value of Y given different values of X here is computed as:

P(Y = 0 | X = 1) = 0.5
P(Y = 0 | X = 2) = 0.5² = 0.25
P(Y = 0 | X = 3) = 0.5³ = 0.125
P(Y = 0 | X = 4) = 0.5⁴ = 0.0625

Also for a fair 4 sided die, we get:
P(X = 1) = P(X = 2) = P(X = 3) = P(X = 4) = 0.25

Therefore, using law of total probability, we get here:
P(Y = 0) = P(Y = 0 |X = 1)P(X = 1) + P(Y = 0 | X = 2)P(X = 2) + P(Y = 0 | X = 3)P(X = 3) + P(Y = 0 | X = 4)P(X = 4)
P(Y = 0) = 0.25*(0.5 + 0.25 + 0.125 + 0.0625) = 0.234375

Now P(X >= 3 | Y = 0) is computed using Bayes theorem here as:
P(X >= 3 | Y = 0) = P(Y = 0 | X = 3)P(X = 3) + P(Y = 0 | X = 4)P(X = 4) / P(Y = 0)

= 0.25*(0.125 + 0.0625) / 0.234375

= 0.2

Therefore 0.2 is the required probability here

b) The value of Y given different values of X here is computed as:

P(Y = 2 | X = 1) = 0
P(Y = 2 | X = 2) = 0.5² = 0.25
P(Y = 2 | X = 3) = 3*0.5³ = 0.375
P(Y = 2 | X = 4) = (4c2)*0.5⁴ = 0.375

The expected value here is computed as:

E(X | Y = 2) = 0*P(Y = 2 | X = 1) + 2*P(Y = 2 | X = 2) + 3*P(Y = 2 | X = 3) + 4*P(Y = 2 | X = 4)

= 0*1 + 0.25*2 + 0.375*3 + 0.375*4

= 3.225

Therefore 3.225 is the expected value of X given Y = 2 here.