finite Math Question

2014-10-15 8:18 pm

Let E be the event that a student decides to study for his final exams and F be the
event that a student decides to party every night on campus, with Pr[E] = 0.71 and Pr[E U F^c ]= 0.84 . What is the value of Pr[F] if E and F are mutually exclusive
events?

回答 (1)

知足常樂

2014-10-15 8:29 pm

✔ 最佳答案

If E and F are mutually exclusive, then
Pr(E ∩ F) = 0
Pr(E ∪ F) = Pr(E) + Pr(F)

Note that for any events A and B,
A = (A ∩ B)∪(A ∩ B')
Pr(A) = Pr(A ∩ B) + Pr(A ∩ B')

It is given that Pr(E) = 0.71.
Also, Pr(E ∪F') = 0.84.

Consider
Pr(E ∪ F') = Pr(E) + Pr(F') - Pr(E ∩ F')
Pr(E ∪ F') = Pr(E) + Pr(F') - [Pr(E) - Pr(E ∩ F)]
Pr(E ∪ F') = Pr(E) + Pr(F') - [Pr(E) - 0]
Pr(E ∪ F') = Pr(E) + Pr(F') - Pr(E)
Pr(E ∪ F') = Pr(F')

Therefore,
Pr(F') = 0.84
1 - Pr(F) = 0.84
Pr(F) = 1 - 0.84 = 0.16

2014-10-15 12:48:09 補充：
同學，題目詢問的是 Pr(F) 那是一個 probability，必在０和１之間。

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