Question

McDonald’s decides to give a Pokemon toy with every Happy Meal. Each time you buy a Happy Meal, you are equally likely to get any one of the 6 types of Pokemon. What is the expected number of Happy Meals that you have to buy until you are able to collect all 6 toys? Answer to 3 decimal places. (Hint: Express this complicated random variable as a sum of geometric random variables, and use linearity of expectation.)

Answer 1

The no of Happy meals until the first Pokemon toy appears is1. After that, the random no of Happy meals until a second (different) toy appears is geometrically distributed with parameter of success 5/6, hence with mean 6/5(recall that the mean of a geometrically distributed random variable is the inverse of its parameter). After that, the random no of Happy meals until a third (different) toy appears is geometrically distributed with parameter of success 4/6, hence with mean 6/4. And so on, until the random no of Happy meals of appearance of the last and sixth toy, which is geometrically distributed with parameter of success 1/6, hence with mean 6/1. This shows that the mean total no of Happy meals to get all six toys is : (1 + 6/5 + 6/4 + 6/3 + 6/2 + 6/1) = 14.70