- A biased coin has probability 0.4 of showing a head. In an experiment, the coin is spun until a head appears. If a head has not appeared after 4 spins, the coin is not spun again. The random variable \(X\) represents the number of times the coin is spun.
For example, \(X = 3\) if the first two spins do not show a head but the third spin does show a head. The coin would not then be spun a fourth time since the coin has already shown a head.
- Show that \(\mathrm { P } ( X = 3 ) = 0.144\)
The table gives some values for the probability distribution of \(X\)
| \(x\) | 1 | 2 | 3 | 4 |
| \(\mathrm { P } ( X = x )\) | | 0.24 | 0.144 | |
- Write down the value of \(\mathrm { P } ( X = 1 )\)
- Find \(\mathrm { P } ( X = 4 )\)
- Find \(\mathrm { E } ( X )\)
- Find \(\operatorname { Var } ( X )\)
The random variable \(H\) represents the number of heads obtained when the coin is spun in the experiment.
- Explain why \(H\) can only take the values 0 and 1 and find the probability distribution of \(H\).
- Write down the value of
- \(\mathrm { P } ( \{ X = 3 \} \cap \{ H = 0 \} )\)
- \(\mathrm { P } ( \{ X = 4 \} \cap \{ H = 0 \} )\)
The random variable \(S = X + H\)
- Find the probability distribution of \(S\)