Question

The probability that house sales will increase in the next 6 months is estimated to be 0.25. The probability that the interest rates on housing loans will go up in the same period is estimated to be 0.74. The probability that house sales or interest rates will go up during the next 6 months is estimated to be 0.89. 1) The probability that house sales will increase but interest rates will not is 2) The events increase in house sales and increase in interest rates in the next 6 months are a) independent. b) mutually exclusive. c) have a probability =1. d) None of the above. 3. The events increase in house sales and no increase in house sales are a) independent. b) mutually exclusive. c) have a probability =1. d) (b) and (c)

Answer 1

Solution :

Let us define some events as follows :

A : The house sales will increase in the next 6 months

B : The interest rates on housing loans will go up in the same period (6 month period)

Then,

(A ∪ B) : The house sales or interest rates will go up during the next 6 months

Now, we have following informations :

P(A) = 0.25, P(B) = 0.74 and P(A ∪ B) = 0.89

1) We have to obtain .

P(A ∪ B) = 0.89

P(A) + P(B) - P(A ∩ B) = 0.89

0.25 + 0.74 - P(A ∩ B) = 0.89

0.99 - 0.89 = P(A  ∩ B)

P(A ∩ B) = 0.10

Hence, the probability probability that house sales will increase but interest rates will not is 0.15.

2) We have to obtain whether the two events A and B are mutually exclusive or independent or have the probability equal to 1 or neither of them holds.

If two events A and B are independent then,

P(A ∩ B) = P(A).P(B)

From part (1) we have, P(A ∩ B) = 0.10

P(A).P(B) = 0.25 × 0.74 = 0.185

Hence, the events increase in house sales and increase in interest rates in the next 6 months are not independent.

If two events are mutually exclusive, then P(A ∩ B) = 0

But we have, P(A ∩ B) = 0.10

Hence, the events increase in house sales and increase in interest rates in the next 6 months are not mutually exclusive.

The events increase in house sales and increase in interest rates in the next 6 months have probability = 0.10

Hence, option (d) is correct.

c) We have to obtain whether the two events A and are mutually exclusive or independent or have the probability equal to 1 or neither of them holds.

If two events A and are independent then,

P(A ∩ ) = P(A).P()

P(A ∩ ) = 0

P(A).P() = 0.25 × (1 - 0.25) = 0.1875

Hence, the events increase in house sales and not increase in house sales are not independent.

If two events A and are mutually exclusive, then

we have,

Hence, the events increase in house sales and not increase in house sales are not mutually exclusive.

Hence, option (b) is correct.