2024 RI P2 Q11

Two brothers Kai and Leo are duathlon (running and cycling) athletes who train regularly. For each day, the probability that Kai cycles is \(\frac{3}{4}\), the probability that he runs is \(\frac{3}{5}\) and the probability he does both is \(p\).

1. Write down, in terms of \(p\), the probability that, on one day, Kai either runs or cycles but not both.[1]
2. Find the range of possible values of \(p\).[2]

On average, Leo cycles \(5\) out of \(7\) days in a week. The probability that Leo cycles when Kai cycles is \(0.9\).

3. Find the probability that Leo cycles when Kai does not.[3]
4. State, in context, two assumptions needed for the number of days Leo cycles over a period of \(5\) weeks to be well modelled by a binomial distribution.[2]

Assume now that the number of days Leo cycles in \(5\) weeks has a binomial distribution.

5. Find the probability that, in the \(5\) weeks, Leo cycles on at least \(20\) days but fewer than \(30\) days.[2]
6. Find the probability that, in the \(5\) weeks, there are exactly \(5\) days in which both brothers do not cycle and they both cycle on the other days.[2]