2024 DHS H1 Q9

At a funfair, Alice plays a game by tossing a biased coin \(100\) times using a computer simulation. Let \(Y\) be the random variable denoting the number of heads obtained and \(p\) be the probability of obtaining a head. It is given that \(100 + \mathrm{E}(Y) = 6 \mathrm{Var}(Y)\).

1. Show that \(p = \frac{1}{3}\).[3]
2. Find the probability of Alice obtaining at least \(40\) heads in the game, given that she had obtained all heads in her first \(10\) tosses.[3]
3. Alice plays \(50\) rounds of the game. Estimate the probability that the mean number of heads Alice obtained in each round is less than \(32\).[3]