The geometric distributionEdexcel A-Level Further Maths: Mind map

Exam questions on The geometric distribution

1. A machine produces components, each of which is defective with probability $0.08$, independently of all the others. The components are inspected one at a time. Let $X$ be the number of components inspected up to and including the first defective one.
   Find the smallest number $n$ of inspections for which the probability that the first defective component has been found by the $n$th inspection exceeds $0.5$.2 marks
2. A driving test candidate passes at each attempt with probability $0.35$, independently of any other attempt. Let $X$ be the number of attempts the candidate makes up to and including the first one that they pass.
   Find the probability that the candidate needs at least $5$ attempts to pass, and the expected number of attempts needed.2 marks
3. An archer hits the target on each shot with probability $p$, independently of all other shots. Let $X$ be the number of shots up to and including the first hit. It is known that $\mathrm{E}(X)=5$.
   Find the value of $p$, and the variance and standard deviation of $X$.3 marks