Biathlon

Problem

In the winter sport biathlon, during each shooting round, five targets have to be hit. In the course of an individual race, a biathlet executes a shooting round by shooting on each target once. This shooting round is modeled by a Bernoulli chain of length 5 with a probablity \(p\) to score a hit.

1. Give an expressions for the following events A and B which described the probability for the event as a function of \(p\).

A: „The biathlete scores exactly four hits.“

B: „The biathlete scores a hit only for the first two shots.“

2. Explain by way of example why modeling a shooting round by means of a Bernoulli chain might disagree with reality.

Solution of part a

We start by considering the probability for event B. Since the probability of a hit is given by math:p, the probability for a miss equals \(1-p\). Correspondingly, the probability for scoring a hit for exactly the first two shots is obtained as \(p^2(1-p)^3\). We check this statement by means of simulation. However, we should not expect perfect agreement.

Let us now consider event A. In analogy to the previous consideration, the probability for a given sequence of four hits and and one miss equals \(p^4(1-p)\). However, the shot which misses is not fixed. The number of possibilites to distribute \(M\) events on \(N\) positions is given by the binomial coefficient

\[\binom{N}{M} = \frac{N!}{M!(N-M)!}.\]

In our case, the desired probability is obtained as

\[\binom{5}{4}p^4(1-p) = 5p^4(1-p).\]

After briefly verifying the binomial coefficient of which we make use here

we once more check our result for the probability by means of a simulation:

Solution of part b

The Bernoulli chain assumes that the probability of a hit is the same for each shot. However, in reality the probability of a hit might for example decrease after a miss.