Question

4 fair 10-sided dice are rolled.

(a)	Find the conditional probability that at least one die lands on 3 given that all 4 dice land on different numbers.

(b)	True or False: If X is the sum of the 4 numbers from one roll, and Y is the maximum of the 4 numbers from one roll, then X and Y are independent random variables.

Answer 1

a) The conditional probability that at least one die lands on 3 given that all 4 dice land on different numbers is computed here as:

= 1 - conditional probability that there is no 3 given that all 4 dice land on different numbers.

therefore 0.4 is the required probability here.

b) Here, we will compute some probabilities as:

P(X = 4) = Probability that sum is 4 which is only possible if we have only 1.
P(X = 4) = (1/10)4

Also P(Y = 1) = Probability that max value is 1 which is only possible if all are 1.
P(Y = 1) = (1/10)4

Also P(X = 4, Y = 1) = (1/10)4 but this is not equal to P(X = 4)P(Y = 1) which means X and Y are not independent.

therefore X and Y are not independent variables here. and the given statement is false here.