Probability of Same Number of Heads and 3/5 Outcomes in 10 Flips and 15 Rolls

[aeubz]

Homework Statement

You have a fair coin. Your friend has a fair die you flip your coin 10 times. Your friend rolls his die 15 times. What is the probability that you get "heads" exactly the same number of times your friend gets "3" or "5". Leave your answer in the form of a summation. No need to simplify! Assume independence here.

Homework Equations

kCn (P)^n (1-P)^k-n

The Attempt at a Solution

10Cn (1/2)^n (1/2)^10-n (coin)
15Cn (1/3)^n (2/3)^15-n (dice)

Sum (n = 1 to 10) of 10Cn (1/2)^n (1/2)^10-n * 15Cn (1/3)^n (2/3)^15-n

 

[vacuum]

This summation represents the probability of getting "heads" the same number of times as getting a "3" or "5" on the die for each of the 10 flips and 15 rolls. We use the combination formula to account for the possible ways of getting the same number of "heads" and "3"/"5" outcomes, and we multiply the probabilities since we are assuming independence between the coin and the die. We sum this over all possible values of n from 1 to 10, representing the number of times we get "heads" and "3"/"5" outcomes. The final answer does not need to be simplified as per the instructions.