Porisma

Worked solution

Score three draws without replacement

In a game, 3 red balls and 7 white balls are placed in a bag. A player draws 3 balls at random and without replacement. The player scores 2 points for every red ball that is drawn, and 1 point for every white ball that is drawn. The total score X is obtained by adding up the scores of the 3 balls.

Find the probability distribution of X. marks

Find E(X) and show that Var(X)=0.49. marks

[2017/PJC/J2MYE/II/7]

CJC · Review problems · Question 1

Answer

P(X=3,4,5,6)=(7/24,21/40,7/40,1/120); E(X)=3.9; Var(X)=0.49.

Possible scores and the combination method

Score and coloursP(X=x)P(X=x)
3 (3W)(73)/(103)=7/24\binom73/\binom{10}3=7/24
4 (2W1R)(72)(31)/(103)=21/40\binom72\binom31/\binom{10}3=21/40
5 (1W2R)(71)(32)/(103)=7/40\binom71\binom32/\binom{10}3=7/40
6 (3R)(33)/(103)=1/120\binom33/\binom{10}3=1/120

The four source columns are displayed as rows; all compositions and probabilities are unchanged.

How did this part go?

Alternative: sequential draws and orderings

The source also gives P(X=3)=(7/10)(6/9)(5/8)=7/24P(X=3)=(7/10)(6/9)(5/8)=7/24; P(X=4)=(7/10)(6/9)(3/8)(3!/2!)=21/40P(X=4)=(7/10)(6/9)(3/8)(3!/2!)=21/40; P(X=5)=(7/10)(3/9)(2/8)(3!/2!)=7/40P(X=5)=(7/10)(3/9)(2/8)(3!/2!)=7/40; and P(X=6)=(3/10)(2/9)(1/8)=1/120P(X=6)=(3/10)(2/9)(1/8)=1/120. The factor 3 counts the position of the single differently coloured ball.

How did this part go?

Calculate the moments

E(X)=3724+42140+5740+61120=3.9.\begin{aligned}&E(X)\\ & =3\frac7{24}+4\frac{21}{40}+5\frac7{40}\\ &{}+6\frac1{120}\\ & =3.9.\end{aligned}

Also

E(X2)=9724+162140+25740+361120=15.7.\begin{aligned}&E(X^2)\\ & =9\frac7{24}+16\frac{21}{40}+25\frac7{40}\\ &{}+36\frac1{120}\\ & =15.7.\end{aligned}

Therefore Var⁡(X)=15.7−(3.9)2=0.49\operatorname{Var}(X)=15.7-(3.9)^2=0.49, as required.

How did this part go?

Common mistakes

Without replacement, successive colour probabilities change; this is not a binomial sample.

Solution notes

Complete distribution and both printed probability methods retained for every score, followed by both moments and variance.

Read the hint