Question

The biathlon is a winter sport that combines cross-country skiing and rifle shooting. For this problem, consider an event with five skiing phases split up by four round of rifle shooting (i.e., ski, shoot, ski, shoot, ski, shoot, ski, shoot, ski). In each round of rifle shooting, skiers must shoot five bullets in an attempt to hit five targets. For each miss, they must ski a small penalty loop.Consider a biathlete with an 82% success rate hitting the target. Assume each shot is independent of the other shots.a) What is the probability that the biathlete hits all five targets in one round of rifle shooting?b) The biathlete figures that if she only misses (i.e., fails to hit) two shots or less for the entire race (i.e., all four rounds of rifle shooting with five shots each round), then she has a chance of winning the race. What is the probability that this biathlete misses two shots or less in a race?The worst that this biathlete has ever done for a whole race is making exactly thirteen shots over the course of the entire race.c) What is the probability that the biathlete ties that worst performance in her next race?d) What is the probability that the biathlete breaks her record for worst-ever shooting performance in her next race?

Answer 1

a) The binomial distribution formula is

`\begin{gathered} P(k)=(nbinomialk)p^k(1-p)^(n-k) \\ k\rightarrow\text{ number of successful trials} \\ n\rightarrow\text{ total number of trials} \\ p\rightarrow\text{ probability of successful trial} \end{gathered}`

Therefore, in our case, since we need the probability of hitting 5 targets with 5 bullets,

`\begin{gathered} P(5)=(5binomial5)(0.85)^5(1-0.85)^(5-5)=1*(0.85)^5(0.15)^0=(0.85)^2 \\ \Rightarrow P(5)=(0.85)^5=0.4437... \end{gathered}`

The exact probability of event a) is (0.85)^5, approximately 0.4437