Answer to Question #168329 in Statistics and Probability for Bikram Chaudhary

Question

If there is an increase in capital investment next year, the probability that the price of structural steel will increase is 0.90. If there is no increase in such investment, the probability of an increase is 0.40. Overall, we estimate that there is 60% chance that the capital investment will increase next year.

i) What is the overall probability of an increase in structural steel prices next year?

ii) Suppose that during the next year structural steel prices in fact increase what is the probability that there was an increase in capital investment?

Answer 1

If there is an increase in capital investment next year, the probability that the price of structural steel will increase is 0.90. If there is no increase in such investment, the probability of an increase is 0.40. Overall, we estimate that there is 60% chance that the capital investment will increase next year.

i) What is the overall probability of an increase in structural steel prices next year?

ii) Suppose that during the next year structural steel prices in fact increase what is the probability that there was an increase in capital investment?

Let I denote the probability that the capital investment will increase next year.

Let S denote the probability that the price of structural steel will increase.

Therefore we have:

$P(I) = 0.6$

$P(S|I)=0.9$

$P(S|I')=0.4$

I) Need to find $P(S)$

$P(S)=P(S|I)P(I)+P(S|I')P(I')$

where $P(I')=1-P(I)=1-0.6=0.4$

thus

$P(S)=0.9\cdot0.6+0.4\cdot0.4=0.7$

II) Need to find $P(I|S)$

Using Bayes' theorem:

$$P(I|S)=\frac{P(S|I)P(I)}{P(S)}=\frac{0.9\cdot0.6}{0.7}=0.77$$

Answer:

I) 0.7

II) 0.77