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Question: Hello! Showing work would be super helpful. I got these questions wrong on a homework assignment ...

Hello! Showing work would be super helpful. I got these questions wrong on a homework assignment because I am not fully understanding the math behind it. Thanks you very much in advance!

This weekend at a certain social club, there is scheduled a pool tournament and a table tennis tournament. It is known that 51.27% of the members are competing in the pool tournament, 41.14% of the members are competing in the table tennis tournament, and that 10.13% are competing for both.

1.) Draw a contingency Table

Suppose you visit the social club and meet a random member find the probability that:

2.) This member is competing in a tournament (either pool or table tennis)

3.) This member is not competing in the table tennis tournament.

4.) This member is competing in the pool tournament but not in the tennis tournament.

5.) This member is not competing in any of the tournaments.

6.) This member is competing in the pool tournament, given that this member is competing in the table tennis tournament.

7.) This member is competing in the table tennis tournament, given that this member is competing in the pool tournament.

8.) Are these two events (competing in the pool tournament and competing in the table tennis tournament) disjoint? Prove or disprove with numbers!

9.) Are these two events independant? Prove or disprove with numbers!

Explanation / Answer

1)

Contingency Table

P(P) = 0.5127
P(T) = 0.4114
P(P and T) = 0.1013

2)
P(P OR T) = P(P) + P(T) - P(P and T)
= 0.5127 + 0.4114 - 0.1013
= 0.8228

3)
P(T') = 1 - P(T) = 1 - 0.4114 = 0.5886

4)
P(P and T') = 0.5127 - 0.1013 = 0.4114

5)
P(P' and T') = 1 - 0.8228 = 0.1772

6)
P(P|T) = P(P and T)/P(T) = 0.1013/0.4114 = 0.2462

7)
P(T|P) = P(P and T)/P(P) = 0.1013/0.5127 = 0.1976

Pool tournament Tennis Both % 51.27% 41.14% 10.13% probability 0.51 0.41 0.10