1An analyst thinks that next year there is a 70 chance that

1)An analyst thinks that next year there is a 70% chance that the world economy will be good, a 10% chance that it will be neutral, and a 20% chance that it will be poor. She also predicts probabilities that the performance of a start-up firm, Creative Ideas, will be good, neutral, or poor for each of the economic states of the world economy. The following table presents probabilities for three states of the world economy and the corresponding conditional probabilities for Creative Ideas.

What is the probability that the performance of the world economy will be neutral and that of creative ideas will be poor?

What is the probability that the performance of Creative Ideas will be poor?

The performance of Creative Ideas was poor. What is the probability that the performance of the world economy had also been poor?

Please show all work so I am able to understand how to solve.

3) The State Police are trying to crack down on speeding on a particular portion of the Massachusetts Turnpike. To aid in this pursuit, they have purchased a new radar gun that promises greater consistency and reliability. Specifically, the gun advertises

Solution

a)
performance of wold economy is neutral ; probability = 0.1
performance of creative ideas is poor; probability = 0.1
so combined probability = 0.1 * 0.1 = 0.01
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b)
case 1)
world economy good ; probability =0.7
creative ideas poor ; probability = 0.3
combined = 0.7 * 0.3 = 0.21

case 2)
world economy neutral ; probability = 0.1
creative ideas poor = 0.1
combined = 0.1 * 0.1 = 0.01

case 3)
world economy poor ; probability = 0.2
creative ideas poor = 0.1
combined = 0.2 * 0.1 = 0.02

net probability that the performance of creative
ideas is poor = 0.21 + 0.01 + 0.02 = 0.24
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c) given that the performance of creative ideas is poor

ie., 0.24 ( we found out in previous one)

case 3) in this case performance of world economy
was also poor
we got the combined probability as 0.02

now required probability is 0.02/0.24 = 1/12 =0.0833

3)
probability that the gun detect a speeder = 0.81
probability for not detecting a speeder will be = 1 - 0.81 = 0.19

probability that the gun erroneously detects a driver
even when the driver is below speed limit = 0.01
other 0.99 is the probability that when the driver is under the
speed limit it won\'t point him as a speeder.

probability that a driver is under speed limit is 0.78
probability that a driver crossing speed limit is 1 - 0.78 = 0.22

a)
driver speeding probability = 0.22
gun detecting probability = 0.81
net required probability = 0.22 * 0.81 = 0.1782

b) driver not speeding = 0.78
gun false detection= 0.01
net required probability = 0.78 * 0.01 = 0.0078

c) in question a and b the gun detects speeding
in part a) driver speed is above speed limit
and in part b) driver speed is below speed limit

so net probability that the gun detecting is
0.1782 + 0.0078 = 0.186

out of this 0.0078 is the probability part b)
where the driver is below speed limit

now required probability is 0.0078/(0.1782 + 0.0078)
= 0.0078/0.186 = 0.0419
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4) P(A|B) = P(A and B)/P(B)

and the event A is said to be independent of B
if P(A|B) = P(A)

a)
here P(A and B) = 0.18
P(B) = P(A) = 0.4
P(A|B) = 0.18/0.4 = 0.45 which is not equal to P(A)
therefore they are not independent events.
so the answer is third option

b)Two events are mutually exclusive
if they cannot occur at the same time.
that means P(A and B) should be 0.
but here P(A and B) = 0.18
therefore they are not mutually exclusive
so the answer is fourth option

c) P( A or B) = P(A) + P(B) - P( A and B)

P(A or B) = 0.4 + 0.4 - 0.18 =0.62

probability that neither A nor B occurs is

1 - 0.62 = 0.38