Bayes' theorem problem, Struggling with this the whole night, please help.Thank you.
Two states of nature exist for a particular situation: a good economy and a poor economy. An economic study may be performed to obtain more information about which of these will actually occur in the coming year. The study may forecast either a good economy or a poor economy. Currently there is 60% chance that the economy will be good and a 40% chance that the economy will be poor. In the past, whenever the economy was good, the economic study predicted it would be good 80% of the time. (The other 20% of the time the prediction was wrong.) In the past, whenever the economy was poor, the economic study predicted it would be 90% of the time.(The other 10% of the time the prediction was wrong.)
a) Use Bayes' theorem and find the following:
P(good economy| prediction of good economy)
P(poor economy| prediction of good economy)
P(good economy| prediction of poor economy)
P(poor economy| prediction of poor economy)
b) Suppose the initial (prior) probability of a good economy is 70% (instead of 60%), and the probability of a poor economy is 30% (instead of 40%). Find the posterior probabilities in part a based on these new values.
Let's solve this step-by-step.
a) Use Bayes' theorem to find the following probabilities:
P(good economy| prediction of good economy)
P(poor economy| prediction of good economy)
P(good economy| prediction of poor economy)
P(poor economy| prediction of poor economy)
First, we need to calculate the prior probabilities.
P(good economy) = 0.6
P(poor economy) = 0.4
Now, let's calculate the conditional probabilities.
P(prediction of good economy| good economy) = 0.8
P(prediction of poor economy| good economy) = 0.2
P(prediction of good economy| poor economy) = 0.9
P(prediction of poor economy| poor economy) = 0.1
We can now apply Bayes' theorem:
P(good economy| prediction of good economy) = (P(prediction of good economy| good economy) * P(good economy)) / (P(prediction of good economy| good economy) * P(good economy) + P(prediction of good economy| poor economy) * P(poor economy))
P(good economy| prediction of good economy) = (0.8 * 0.6) / (0.8 * 0.6 + 0.9 * 0.4)
P(good economy| prediction of good economy) = 0.8
P(poor economy| prediction of good economy) = 1 - P(good economy| prediction of good economy)
P(poor economy| prediction of good economy) = 1 - 0.8
P(poor economy| prediction of good economy) = 0.2
P(good economy| prediction of poor economy) = (P(prediction of poor economy| good economy) * P(good economy)) / (P(prediction of poor economy| good economy) * P(good economy) + P(prediction of poor economy| poor economy) * P(poor economy))
P(good economy| prediction of poor economy) = (0.2 * 0.6) / (0.2 * 0.6 + 0.1 * 0.4)
P(good economy| prediction of poor economy) = 0.4615 (rounded to 4 decimal places)
P(poor economy| prediction of poor economy) = 1 - P(good economy| prediction of poor economy)
P(poor economy| prediction of poor economy) = 1 - 0.4615
P(poor economy| prediction of poor economy) = 0.5385 (rounded to 4 decimal places)
b) Suppose the initial (prior) probability of a good economy is 70% and the probability of a poor economy is 30%. Calculate the posterior probabilities in part a based on these new values.
Using the same conditional probabilities calculated in part a:
P(good economy| prediction of good economy) = (0.8 * 0.7) / (0.8 * 0.7 + 0.9 * 0.3)
P(good economy| prediction of good economy) = 0.8485 (rounded to 4 decimal places)
P(poor economy| prediction of good economy) = 1 - P(good economy| prediction of good economy)
P(poor economy| prediction of good economy) = 1 - 0.8485
P(poor economy| prediction of good economy) = 0.1515 (rounded to 4 decimal places)
P(good economy| prediction of poor economy) = (0.2 * 0.7) / (0.2 * 0.7 + 0.1 * 0.3)
P(good economy| prediction of poor economy) = 0.7
P(poor economy| prediction of poor economy) = 1 - P(good economy| prediction of poor economy)
P(poor economy| prediction of poor economy) = 1 - 0.7
P(poor economy| prediction of poor economy) = 0.3
Therefore, the updated posterior probabilities in part a, considering the new initial probabilities, are as follows:
P(good economy| prediction of good economy) = 0.8485
P(poor economy| prediction of good economy) = 0.1515
P(good economy| prediction of poor economy) = 0.7
P(poor economy| prediction of poor economy) = 0.3
To solve this problem using Bayes' theorem, we need to understand the given probabilities and use them to calculate the posterior probabilities. Bayes' theorem provides a way to update our prior beliefs (prior probabilities) based on new evidence (conditional probabilities).
a) Using the given probabilities:
Let's define the events:
G = good economy
P = poor economy
E = prediction of good economy
~E = prediction of poor economy
We are given the following probabilities:
P(G) = 0.6 (prior probability of a good economy)
P(P) = 0.4 (prior probability of a poor economy)
P(E|G) = 0.8 (probability of predicting good economy given the economy is good)
P(~E|G) = 0.2 (probability of predicting poor economy given the economy is good)
P(E|P) = 0.9 (probability of predicting good economy given the economy is poor)
P(~E|P) = 0.1 (probability of predicting poor economy given the economy is poor)
Now, let's use Bayes' theorem:
1) P(G|E) = (P(E|G) * P(G)) / P(E)
P(E) = P(E|G) * P(G) + P(E|P) * P(P) [The law of total probability]
P(G|E) = (P(E|G) * P(G)) / P(E|G) * P(G) + P(E|P) * P(P)
P(G|E) = (0.8 * 0.6) / (0.8 * 0.6 + 0.9 * 0.4)
2) P(P|E) = 1 - P(G|E) [Since there are only two states]
3) P(G|~E) = (P(~E|G) * P(G)) / P(~E)
P(~E) = P(~E|G) * P(G) + P(~E|P) * P(P)
P(G|~E) = (P(~E|G) * P(G)) / P(~E|G) * P(G) + P(~E|P) * P(P)
P(G|~E) = (0.2 * 0.6) / (0.2 * 0.6 + 0.1 * 0.4)
4) P(P|~E) = 1 - P(G|~E)
b) If the initial (prior) probability of a good economy is 70% and the probability of a poor economy is 30%, we need to update our calculations using these new values and follow the same procedure explained above.
P(G) = 0.7 (updated prior probability of a good economy)
P(P) = 0.3 (updated prior probability of a poor economy)
Repeat steps 1 to 4 using the updated values to calculate the posterior probabilities P(G|E), P(P|E), P(G|~E), and P(P|~E).
By applying Bayes' theorem and using the given probabilities, we can calculate the posterior probabilities in part a. To calculate the posterior probabilities in part b, we use the new prior probabilities and repeat the same steps.