Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 7 / c / ii

Table of contents
- Solution ii

Solution

Let

B

count the red balls among the

n

added balls. By independence of random variables,

B

has a binomial distribution with parameters

n, 1/2

. The total red count is

R = 1 + B

, and the box has

n + 1

balls. Conditional on

R = r

, sampling without replacement gives

P (D_{1} ∣ R = r) = \frac{r}{n + 1}, P (D_{1} \cap D_{2} ∣ R = r) = \frac{r ( r - 1 )}{n ( n + 1 )},

where

D_{1}, D_{2}

denote red first and second draws. Averaging these conditional probabilities and taking their ratio gives

P (D_{2} ∣ D_{1}) = \frac{E [ R ( R - 1 )]}{n E [ R ]} .

The binomial distribution moments give

E [B] = n /2

and

E [B^{2}] = n /4 + n^{2} /4

. Consequently

E [R] = \frac{n + 2}{2}, E [R (R - 1)] = E [B^{2} + B] = \frac{n ( n + 3 )}{4} .

Therefore the red-ball prediction with one certain red ball is

P (D_{2} ∣ D_{1}) = \frac{n + 3}{2 ( n + 2 )}, n \geq 1.

It reduces to

2/3

at

n = 1

and approaches

1/2

as the number of randomly coloured extra balls grows. Taking an unweighted average of the remaining red fractions would miss the change in weights caused by taking a conditional probability given the first red draw.

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