Discrete Random Variables (DRV)

A bag contains \(2\) red balls, \(3\) yellow balls and \(1\) blue ball. Temmy and Ryan play a game where each takes turns to draw a ball at random from the bag, with replacement. The number of red balls obtained in \(n\) fixed draws from the bag is denoted by \(R\).

1. State, in context, an assumption satisfied by \(R\) for it to be well modelled by a binomial distribution.
2. Temmy and Ryan each draw \(n\) times from the bag. Find the least \(n\) such that the probability of both getting a total of at most \(10\) red balls is not more than \(0.5\).
3. Temmy and Ryan each draw \(5\) times from the bag. The player with more red balls drawn wins. Otherwise, the game ends in a draw. Find the probability that Temmy wins the game if she draws more than \(3\) red balls.

In a variation of the game, Temmy draws balls at random from the bag, one at a time without replacement, and stops when she obtains \(2\) yellow balls. The total number of balls Temmy has to draw from the bag before she stops is denoted by \(T\). Find \(\mathrm{E}(T)\) and \( \mathrm{Var}(T)\).