A standard balanced 6 sided die is rolled is rolled 10 times.
a. Find the probability that five of the rolls are odd numbers and five of the rolls are even numbers.
P(odd) = (10 choose 5)(1/2)^(5)(1/2)^(5) = P(even)
b. Find the expected value and variance of the number of threes rolled.
Let Ii be an indicator variable indicating whether a 3 is rolled. Then E[I1 +...i10] = 10(1/6) = 10/6.
c. Find the probability that exactly two sixes are rolled given five twos are rolled.
These are independent events so P(2 6's rolled|5 2's rolled) = P(2 6's rolled and 5 2's rolled)/P(5 2's rolled) = (10 choose 2)(1/6)^(2)(5/6)^(8).
Are these answers correct?