Of the 15 teams in the National Hockey League Eastern Conference, three of them are based in canadian cities. Each year, eight of these teams qualify for the playoffs. Assuming that all the 15 teams have an equal chance of getting into the playoffs, what is the probability that
a) none of the canadian teams make the playoffs
b) at least one of the canadian teams makes the playoffs
c) all three canadian teams make the playoffs.
**if i even get hints or help with one part i am sure i can figure it out. thanks.
To calculate the probabilities, we need to first determine the total number of possible outcomes. In this case, there are 15 teams in total, and 8 teams will make it to the playoffs.
a) To find the probability that none of the Canadian teams make the playoffs, we need to determine how many ways we can choose 8 teams from the non-Canadian teams. Since there are 12 non-Canadian teams, we can calculate this using the combination formula:
C(12, 8) = 12! / (8!(12-8)!) = 495
The total number of possibilities is C(15, 8) = 15! / (8!(15-8)!) = 6435
Therefore, the probability that none of the Canadian teams make the playoffs is:
P(a) = 495 / 6435 ≈ 0.0767 or 7.67%
b) To determine the probability that at least one of the Canadian teams makes the playoffs, we can calculate the probability of the complement event, which is the event where none of the Canadian teams make the playoffs.
P(b) = 1 - P(a) ≈ 1 - 0.0767 = 0.9233 or 92.33%
c) To find the probability that all three Canadian teams make the playoffs, we need to determine how many ways we can choose 8 teams from the 12 non-Canadian teams and subtract it from the total number of possibilities.
C(12, 8) = 495 as calculated in part a)
Thus, the probability that all three Canadian teams make the playoffs is:
P(c) = 1 - (495 / 6435) ≈ 0.9233 - 0.0767 = 0.8466 or 84.66%
I hope this explanation helps! Let me know if you have any further questions or need clarification on any steps.
a) None of the Canadian teams make the playoffs:
To find the probability that none of the Canadian teams make the playoffs, we need to calculate the probability that all three Canadian teams do not qualify for the playoffs.
There are three Canadian teams and 12 non-Canadian teams in the conference. Since all teams have an equal chance, the probability that a Canadian team does not make the playoffs is 12/15.
Since there are three Canadian teams, the probability that all three do not make the playoffs is (12/15) * (12/15) * (12/15) = 1728/3375.
b) At least one of the Canadian teams makes the playoffs:
To find the probability that at least one of the Canadian teams makes the playoffs, we can find the complementary probability (the probability that none of the Canadian teams make the playoffs) and subtract it from 1.
The complement of none of the Canadian teams making the playoffs is at least one of the Canadian teams making the playoffs. Therefore, the probability that at least one Canadian team makes the playoffs is 1 - (1728/3375) = 1647/3375.
c) All three Canadian teams make the playoffs:
To find the probability that all three Canadian teams make the playoffs, we need to calculate the probability that each Canadian team qualifies for the playoffs.
The probability that a Canadian team makes the playoffs is 8/15, since 8 teams out of the 15 qualify.
Therefore, the probability that all three Canadian teams make the playoffs is (8/15) * (8/15) * (8/15) = 512/3375.