Here's another tricky probability question.
A soccer player scores an average of 3 goals per game:
a) What is the probability that he or she will score 4 goals in a given game?
To determine the probability of the soccer player scoring exactly 4 goals in a given game, we can use the Poisson distribution. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space, given the average rate of occurrence.
In this case, since the soccer player scores an average of 3 goals per game, the average rate of occurrence is λ = 3.
The probability of getting exactly k goals in a game using the Poisson distribution formula is given by:
P(k, λ) = (e^(-λ) * λ^k) / k!
Where e is the base of the natural logarithm (approximately 2.71828) and k! represents k-factorial (the product of all positive integers less than or equal to k).
Plugging in the values for k = 4 and λ = 3 into the formula, we can calculate the probability:
P(4, 3) = (e^(-3) * 3^4) / 4!
Calculating the numerator:
e^(-3) ≈ 0.0498
3^4 = 81
Calculating the denominator:
4! = 4 * 3 * 2 * 1 = 24
Now we can substitute these values back into the formula:
P(4, 3) = (0.0498 * 81) / 24
P(4, 3) ≈ 0.1665
Therefore, the probability that the soccer player will score exactly 4 goals in a given game is approximately 0.1665 or 16.65%.