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cricket20 [7]
3 years ago
12

The prior probabilities for events A1 and A2 are P(A1) = 0.30 and P(A2) = 0.55. It is also known that P(A1 ∩ A2) = 0. Suppose P(

B | A1) = 0.20 and P(B | A2) = 0.05. If needed, round your answers to three decimal digits.a) Are A1 and A2 mutually exclusive? b) Compute P(A1 ∩ B) and P(A2 ∩ B). c) Compute P(B). d) Apply Bayes’ theorem to compute P(A1 | B) and P(A2 | B).
Mathematics
1 answer:
k0ka [10]3 years ago
3 0

We're given the following probabilities:

P(A_1)=0.30

P(A_2)=0.55

P(A_1\cap A_2)=0

P(B\mid A_1)=0.20

P(B\mid A_2)=0.05

(a) Yes, A_1 and A_2 are mutually exclusive. This is exactly what zero probability of their intersection means. The two events cannot occur simultaneously.

(b) Use the definition of conditional probability to expand:

P(A_1\cap B)=P(A_1)P(B\mid A_1)=0.30\cdot0.20=0.06

P(A_2\cap B)=P(A_2)P(B\mid A_2)=0.55\cdot0.05=0.0275

(c) By the law of total probability,

P(B)=P(A_1\cap B)+P(A_2\cap B)=0.06+0.0275=0.0875

(d) Bayes' theorem says

P(A_1\mid B)=\dfrac{P(A_1)P(B\mid A_1)}{P(B)}=\dfrac{0.30\cdot0.20}{0.0875}\approx0.686

P(A_2\mid B)=\dfrac{P(A_2)P(B\mid A_2)}{P(B)}=\dfrac{0.55\cdot0.05}{0.0875}\approx0.314

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