Practice section 4 of 7

Four distinct cards are numbered 1,2,3,4. A player randomly picks a pair without replacement. If their sum is odd, that sum is the score. If the sum is even, the player randomly picks a third card from the remaining cards and scores the square of its number.

(i) Find the probability of score 4.

(ii) Find the probability distribution of score S, and hence its expectation.

(iii) Find the probability of a score below 5, given three cards are drawn.

An unbiased regular tetrahedral die has faces 1,2,3,4. On throwing it onto a horizontal table, the face in contact with the table gives the number. Two such dice are thrown and X is the product of their numbers. Obtain the probability distribution of X, then derive

(i) $P(X>8)$,

(ii) $E(X)$,

(iii) $\operatorname{Var}(X)$.

Alex and his friend stand randomly in a queue with three other people. X is the number of people between Alex and his friend.

(i) Show that $P(X=2)=0.2$.

(ii) Tabulate the distribution of X.

(iii) Find $E(X)$ and $E[(X-1)^2]$. Hence find $\operatorname{Var}(X)$.

The probability function of X is

$$P(X=x)=\begin{cases}(2x-1)\theta,&x=1,2,3,\\k,&x=4,\\0,&\text{otherwise},\end{cases}$$

where $0<\theta<1/9$.

(i) Show that $k=1-9\theta$. Find the distribution of X in terms of θ.

(ii) Find $E(X)$ in terms of θ and hence show that $\operatorname{Var}(X)=26\theta-196\theta^2$.

(iii) $Y=a+bX$, with a,b non-zero constants. Given $\operatorname{Var}(Y)=b^2/3$, find θ.

A cubical die has three faces marked 1, two faces marked 2 and one face marked 3. Find the expectation and variance of its score in one throw.

Deduce, or find otherwise, the expectation and variance for a second cubical die with one face marked 1, two marked 2 and three marked 3.

Two dice of the first type and one of the second are thrown together; X is their total score. Let $\mu=E(X)$ and $\sigma^2=\operatorname{Var}(X)$. Show that

(i) $\sigma^2=5/3$,

(ii) $P(|X-\mu|>2\sigma)=1/18$.

Two coins are tossed. If they show different faces, the game stops. If they show the same face, both are tossed again together with another coin, and the game stops after that round. X is the total number of heads obtained throughout the game.

(i) Show that $P(X=2)=1/8$.

(ii) Tabulate the probability distribution of X.

(iii) A player pays k dollars to play and receives one dollar for each head obtained. Find k for a fair game.