Coursework questions

Andy and Barry play a darts match. The first to win either two games in a row or three games wins the match.

List the possible outcomes of the match, i.e. list the outcomes in the sample space. It may be helpful to make a tree diagram.

[2]

By using only the axioms of probability, and the properties of sets, prove that if $A\subseteq B$ then $\mathrm{P}(A)\le \mathrm{P}(B)$. Hint: Express $B$ as a disjoint union of $A$ and another set.

[4]

Suppose that among $100,000$ women with negative mammograms, $20$ will get breast cancer and that $1$ woman in $10$ with positive mammograms will get breast cancer. Let $A$ denote positive mammograms, $B$ breast cancer.

(a) Write down $\mathrm{P}(B|A^c)$, and $\mathrm{P}(B|A)$.

(b) Suppose that $7\%$ of the general (female) population will have a positive mammogram. Find the probability of developing breast cancer among the general (female) population.

[4]