statistics, events

Question

Hi!

In a biathlon competition there are four stations where the competitors stop and shot. At each station there are five boards and the competitors have eight attempts to hit five boards. For every board that has not been hit at a station, the competitor need to go a penalty round. A competitor has an 80 percent probability of hitting the board. What is the probability that the competitor may go at least one penalty round?

Answer 1

To find the probability a competitor goes to at least one penalty round, first, let's find the probability they go to zero penalty rounds. Then we can just subtract our result from 1, because the two events are complements ("at least one" is the complement of "none").

First, what is the probability they make every shot at a single station? It's a binomial random variable with n = 8 and p = .8 and they need at least 5 successful shots.

sum i = 5 to 8 of [(8 choose i) (.8)^i (1-.8)^(8-i)] = 0.9437184.

Next, what's the probability that this event (making all 5 shots) occurs at each of the 4 stations? Again, this is a binomial random variable, but this time, n = 4 and p = 0.9437184.

(4 choose 4) (0.9437184)^4 (1-0.9437184)^(4-4) = (0.9437184)^4 ≈ 0.7932

Therefore the probability a competitor goes to at least one penalty round is 1 - 0.7932 = 0.2068.