To find the probability that a randomly chosen NHL player is from Western Canada, we need to use conditional probability.
Let's break down the problem step by step:
1. We are given that 60% of hockey players are from Eastern Canada, which means that the remaining 40% must be from Western Canada.
P(E) = 0.6 (probability of Eastern player)
P(W) = 0.4 (probability of Western player)
2. We are also given that 18% of Eastern players and 12% of Western players make it to the NHL.
P(NHL|E) = 0.18 (probability of being an NHL player given they are from Eastern Canada)
P(NHL|W) = 0.12 (probability of being an NHL player given they are from Western Canada)
3. Now, we want to find the probability that a randomly chosen NHL player is from Western Canada.
We can use Bayes' Theorem to calculate this:
P(W|NHL) = P(NHL|W) * P(W) / P(NHL)
P(NHL) can be calculated by using the Law of Total Probability:
P(NHL) = P(NHL|E) * P(E) + P(NHL|W) * P(W)
Now, let's substitute the given values into the formulas:
P(NHL|W) = 0.12
P(W) = 0.4
P(E) = 0.6
P(NHL|E) = 0.18
P(NHL) = P(NHL|E) * P(E) + P(NHL|W) * P(W)
= 0.18 * 0.6 + 0.12 * 0.4
After calculating P(NHL), we can substitute all the values back into the original formula:
P(W|NHL) = P(NHL|W) * P(W) / P(NHL)
Now, you can plug in the values to calculate the probability that a randomly chosen NHL player is from Western Canada.