s1 HEELPP PLZ I HAVE AN OFFICIAL TMW

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.
(a) Show that P(X = 3) = 0.144

The table gives some values for the probability distribution of X
x = 1234
P(X = x) 2=0.24 3=0.144
(b) (i) Write down the value of P(X = 1)
(ii) Find P(X = 4)

(c) Find E(X )

(d) Find Var(X )

The random variable H represents the number of heads obtained when the coin is spun in
the experiment.

(e) Explain why H can only take the values 0 and 1 and find the probability distribution
of H.

(f) Write down the value of
(i) P({X = 3} ∩ {H = 0})
(ii) P({X = 4} ∩ {H = 0})

The random variable S = X + H
(g) Find the probability distribution of S