Question

The probabilities that stock A will rise in price is 0.50 and that stock B will rise in price is 0.50. Further, if stock B rises in price, the probability that stock A will also rise in price is 0.35.a. What is the probability that at least one of the stocks will rise in price? (Round your answer to 2 decimal places.) Probability b. Are events A and B mutually exclusive?Yes because P(A | B) = P(A).Yes because P(A ∩ B) = 0.No because P(A | B) ≠ P(A).No because P(A ∩ B) ≠ 0.c. Are events A and B independent?Yes because P(A | B) = P(A).Yes because P(A ∩ B) = 0.No because P(A | B) ≠ P(A).No because P(A ∩ B) ≠ 0.

Answer 1

Answer:

a) 0.825 = 0.83 to 2 d.p

b) No because P(A ∩ B) ≠ 0.

c) No because P(A | B) ≠ P(A)

Step-by-step explanation:

P(A) = 0.5

P(B) = 0.5

P(A|B) = 0.35

P(A|B) = P(A n B)/P(B)

P(A n B) = P(A|B) × P(B) = 0.35 × 0.5 = 0.175

a) The probability that at least one of the stocks will rise in price = 1 - (Probability that none of the stocks will rise)

Probability that none of the stocks will rise = P(A' n B')

But the universal probability, P(U) is given by

P(U) = [P(A) - P(A n B)] + [P(B) - P(A n B)] + P(A n B) + P(A' n B')

P(U) = 1

1 = [0.5 - 0.175] + [0.5 - 0.175] + 0.175 + P(A' n B')

P(A' n B') = 1 - 0.175 - 0.325 - 0.325 = 0.175

Probability that at least one of the stocks will rise = P(A n B') + P(A' n B) + P(A n B) = 1 - P(A' n B') = 1 - 0.175 = 0.825

b) Are events A and B mutually exclusive?

For two events to be mutually exclusive, it means the two events cannot occur at the same time (simultaneously). Therefore, they both do not have any similar data points.

Mathematically, it is expressed that for two events A and B to be mutually exclusive,

P(A n B) = 0.

But for this problem, P(A n B) was obtained to be 0.175 ≠ 0, hence the two events are not mutually exclusive.

c) Are events A and B independent?

For two events to be independent, it means the chances or probability of one of them occurring does not in any way depend on the chances of the other one occurring.

The first mathematical test for independent events is that

P(A n B) = P(A) × P(B)

And Mathematically, for two independent events A & B,

P(A|B) = P(A) and P(B|A) = P(B)

For this question, P(A n B) ≠ P(A) × P(B) (0.175≠0.25) and P(A|B) ≠ P(A) (0.35 ≠ 0.5)

Hence, the two events are not independent either.

Hope this helps!!!

Answer 2

Answer:

a. 0.825

b.A and B are not mutually exclusive because P(A ∩ B) ≠ 0

c. A and B are not independent because P(A | B) ≠ P(A)

Step-by-step explanation:

a.As the events A and B are not independent or mutually exclusive, the probability of A and B to happen at the same time is

P(A ∩ B) = P(A|B)P(B) = 0.35 * 0.5 = 0.175

So the probability of either A or B to happen (at least 1 stock rises in price) is

P(A U B) = P(A) + P(B) - P(A ∩ B) = 0.5 + 0.5 - 0.175 = 0.825

b.A and B are not mutually exclusive because P(A ∩ B) ≠ 0

c. A and B are not independent because P(A | B) ≠ P(A)