Practice section 6 of 7

An unbiased ordinary die is thrown. If its score is not 3, the result is that score. If it is 3, the die is thrown again and the result is the absolute difference of the two scores. Let R denote the result. Show that $P(R=0)=1/36$ and $P(R=1)=2/9$, and find the distribution of R. [6]

Find

(i) the mean result, [2]

(ii) the probability that the difference between $(R-1)^2$ and 3 is at most the mean from (i). [4]

S has distribution

s	$P(S=s)$
1	0.5
5	0.3
10	0.1
20	0.1

A circle has radius $R=2S$.

(i) Find $P(S(11-S)\le10)$. [3]

(ii) Find $E(S)$ and $\operatorname{Var}(S)$. [3]

(iii) Hence find the mean and variance of the circle’s area A. [5]

A fairground game involves shooting at a moving target. A round has a maximum of 3 shots and ends immediately after a miss. Each hit scores 10 points. Linda hits with probability 0.6 on each shot, and all shots taken are independent.

(i) Find the probability that she scores 30 points in a round. [2]

X is her score in one round.

(ii) Find its distribution. [3]

(iii) Find its mean and variance. [4]

(iv) A game consists of two rounds. Find the probability that she scores more in round 2 than round 1. [2]

A box contains 2 red balls,3 green balls and x blue balls, with integer $x\ge5$. A contestant chooses 5 balls without replacement. S is the number of green balls plus three times the number of red balls. Blue balls score nothing unless all 5 balls are blue, in which case S=25.

(i) Show that $P(S=6)=\dfrac{20x(x^2-3x+14)}{(x+5)(x+4)(x+3)(x+2)(x+1)}$. [2]

(ii) Given $P(S=6)=5/63$, calculate x. [2]

(iii) Complete the distribution table. [4]

s	Given $P(S=s)$
1	to find
2	$5/42$
3	$5/63$
4	to find
5	$5/21$
6	$5/63$
7	to find
8	$5/84$
9	$1/252$
25	to find

(iv) Evaluate $E(S)$ and find $P(S>E(S))$. [2]

(v) Find the probability that no green balls are drawn, given $S>E(S)$. [2]

Two unbiased regular tetrahedral dice have faces 1,2,3,4. The number on the face in contact with the table is recorded. Tom throws both and notes their total X. Harry guesses the total; his guess is Y. If correct, Harry wins $Y^2$ dollars; otherwise he loses $Y/2$ dollars.

(i) Obtain the distribution of X and find $E(1/X)$ correct to 3 decimal places. [4]

(ii) Harry guesses 6 in a particular game. Find his expected gain. [2]

A board game uses eight-sided dice. The attack die has five faces marked 0, two marked 1 and one marked 2. After the attack throw, the defender throws a defence die with two faces marked 0, four marked 1 and two marked 2. Damage is attack score minus defence score, except that a negative difference gives zero damage.

Let A,D denote the attack and defence scores.

(i) Write down their distributions. Hence find the expectation and variance of $A-D$. [4]

X is the damage in one round.

(ii) Find its distribution, expectation and variance. [5]

(iii) Explain why $E(X)>0$ although $E(A)<E(D)$. [1]