Final answer:
The probability the hockey player will score on his fifth shot is approximately 9.55%. The probability of scoring in fewer than 3 shots is 32.76%. On average, the player should expect to take about 5.56 shots before scoring.
Step-by-step explanation:
The probability of a hockey player scoring on his fifth shot, given he scores on 18% of his shots, can be found using the geometric distribution. We assume that the first four shots were misses (82% chance for each), and the fifth shot was a hit (18% chance).
The probability of scoring on the fifth shot (miss, miss, miss, miss, score) is:
(0.82)^4 × (0.18) approximately equals 0.0955 or 9.55%.
For the probability that it will take him fewer than 3 shots to score (either scoring on the first or second shot), we can add the probability of scoring on the first shot (18%) to the probability of missing the first shot and scoring on the second shot (82% × 18% = 14.76%). So, the probability of scoring in fewer than 3 shots is 18% + 14.76% equals 32.76%.
To calculate the expected number of shots before scoring, which is also known as the mean of a geometric distribution, we use the formula E(X) = 1/p, where p is the probability of success on each shot. Thus, E(X) = 1/0.18 approximately equals 5.56 shots.
This means, on average, the player should expect to take about 5.56 shots before scoring.