In a computer game, a star moves across the screen, with constant speed, taking 1 s to travel from one side to the other. The player can stop the star by pressing a key. The object of the game is to stop the star in the middle of the screen by pressing the key exactly 0.5 s after the star first appears. Given that the player actually presses the key 7 s after the star first appears, a simple model of the game assumes that T is a continuous uniform random variable defined over the interval [0, 1].
- Write down P(T < 0.2). [1]
- Write down E(T). [1]
- Use integration to find Var(T). [4]
A group of 20 children each play this game once.
- Find the probability that no more than 4 children stop the star in less than 0.2 s. [3]
The children are allowed to practise this game so that this continuous uniform model is no longer applicable.
- Explain how you would expect the mean and variance of T to change. [2]
It is found that a more appropriate model of the game when played by experienced players assumes that T has a probability density function g(t) given by
$$g(t) = \begin{cases}
4t, & 0 \leq t \leq 0.5, \\
4 - 4t, & 0.5 \leq t \leq 1, \\
0, & otherwise.
\end{cases}$$
- Using this model show that P(T < 0.2) = 0.08. [2]
A group of 75 experienced players each played this game once.
- Using a suitable approximation, find the probability that more than 7 of them stop the star in less than 0.2 s. [4]