Infogenics, Inc. Stock Prediction: Probability Analysis and Bayes' Theorem

An investment advisor recommends the purchase of shares in Infogenics, Inc. He has made the following predictions:

P(stock goes up 20% | rise in GDP) = 0.6 P(stock goes up 20% | level GDP) = 0.5 P(stock goes up 20% | fall in GDP) = 0.4

An economist has predicted that the probability of a rise in the GDP is 30%, whereas the probability of a fall in the GDP is 40%.

a. Draw a tree diagram to represent this multiple-step experiment. b. What is the probability that the stock will go up 20%? c. We have been informed that the stock has gone up 20%. What is the probability of a rise or fall in the GDP?

a. Here is a tree diagram representing the multiple-step experiment:

                      P(rise in GDP) = 0.3
                          /      \
                         /        \
                        /          \
       P(stock goes up 20%) = 0.6   P(fall in GDP) = 0.4
                |                             |
                |                             |
                |                             |
       P(level GDP) = 0.5             P(stock goes up 20%) = 0.4
                |                             |
                |                             |
                |                             |
     P(stock goes up 20%) = 0.5         P(stock goes up 20%) = 0.3

b. To calculate the probability that the stock will go up 20%, we need to consider all possible paths that lead to this outcome:

P(stock goes up 20%) = P(rise in GDP) * P(stock goes up 20% | rise in GDP)
                       + P(level GDP) * P(stock goes up 20% | level GDP)
                       + P(fall in GDP) * P(stock goes up 20% | fall in GDP)

                     = 0.3 * 0.6 + 0.5 * 0.5 + 0.4 * 0.4
                     = 0.18 + 0.25 + 0.16
                     = 0.59

Therefore, the probability that the stock will go up 20% is 0.59 (or 59%).

c. If we have been informed that the stock has gone up 20%, we need to calculate the probability of a rise or fall in the GDP. This can be determined using Bayes' theorem:

P(rise in GDP | stock goes up 20%) = (P(stock goes up 20% | rise in GDP) * P(rise in GDP)) / P(stock goes up 20%) P(fall in GDP | stock goes up 20%) = (P(stock goes up 20% | fall in GDP) * P(fall in GDP)) / P(stock goes up 20%)

Using the given probabilities:

P(rise in GDP | stock goes up 20%) = (0.6 * 0.3) / 0.59 P(fall in GDP | stock goes up 20%) = (0.4 * 0.4) / 0.59

Calculating these probabilities:

P(rise in GDP | stock goes up 20%) = 0.18 / 0.59 P(fall in GDP | stock goes up 20%) = 0.16 / 0.59

Therefore, the probability of a rise in the GDP given the stock has gone up 20% is approximately 0.305 (or 30.5%), and the probability of a fall in the GDP given the stock has gone up 20% is approximately 0.271 (or 27.1%).