Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 14

Let $N_L$ count successful lemon seeds. By independence it has a binomial distribution with five trials and success probability $y$. Therefore

Take the complement of an event: fewer than two successes means zero or one. These mutually exclusive events have probabilities $(1-y{)}^5$ and $5y(1-y{)}^4$. Hence

Let $N_A$ be the independent avocado count. There are three disjoint ways to have fewer than two total trees: no trees, only one avocado, or only one lemon. Their probabilities giveThe sum is generally a Poisson binomial distribution, rather than a single binomial distribution, because the two success probabilities may differ.

There are three possible count pairs conditional on two trees. Their probabilities, from the two independent binomial distributions, areThe first two correspond to matching kinds, so the conditional probability isThis form remains valid at endpoints whenever the conditioning event has positive probability; if the denominator vanishes, the conditional probability is undefined. As a check, when $x=y\in (0,1)$ the result is $20/45=4/9$, the fraction of two-element subsets containing matching kinds.

Each tree independently has ten equally likely integer yields. Of the $100$ ordered pairs, exactly $(30,32),(31,31),(32,30)$ sum to sixty-two. ThusThe uniform distribution on a finite set applies to the individual yields, not to their sum.

For $n$ independent seeds the probability of at least one success is $1-(1-x{)}^n$. Requiring it to be at least $0.95$ gives $(1-x{)}^n\le 0.05$. For $0<x<1$, taking logarithms and dividing by the negative number $\log (1-x)$ reverses the inequality:The smallest permitted integer is thereforeThe ceiling function is necessary because seeds are counted discretely. If $x=0$, no finite number suffices; if $x=1$, one seed suffices. The target is interpreted as at least a ninety-five percent chance, since exact equality may not be attainable with an integer count.