tmua.me Extreme Set A Paper 2 · Question 8

Each of two bags contains red and blue counters, with at least one counter of each colour in each bag. The first bag contains more red counters than the second bag.

For a real number $p$ with $0<p<1$ , an experiment consists of choosing the first bag with probability $p$ , choosing the second bag otherwise, and then drawing two counters successively from the chosen bag without replacement. Each remaining counter in that bag is equally likely to be drawn.

Let $U(p)$ be the probability that the first counter is red. Let $R(p)$ be the probability that the second counter is red, given that the first is red, and let $B(p)$ be the probability that the second counter is blue, given that the first is blue.

It is given that

R\!\left(\frac13\right)=R\!\left(\frac23\right).

Which statement must be true as $p$ increases with $0<p<1$ ?

U(p)

and

B(p)

are both strictly increasing.

U(p)

and

B(p)

are both strictly decreasing.

U(p)

is strictly increasing and

B(p)

is strictly decreasing.

U(p)

is constant and

B(p)

is strictly increasing.

U(p)

is strictly decreasing and

B(p)

is strictly increasing.

U(p)

is strictly decreasing and

B(p)

is constant.

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tmua.me Extreme Set A Paper 2 · Q8