when it rains, there is a 20% chance that Jamie will go to the gym. When it does not rain, there is a 60% chance that Jamie will go to the gym. there is a 40% chance that it will rain tomorrow.
a. what is the probability that Jamie will go to the gym
b. if Jamie goes to the gym, what is the probability that it rained?
c. if Jamie doesn't go the gym, what is the probability that it rained?
To calculate the probabilities, we can make use of conditional probability.
a. To find the probability that Jamie will go to the gym, we need to consider both cases: when it rains and when it doesn't rain. We can use the Law of Total Probability:
P(Jamie goes to the gym) = P(Jamie goes to the gym | It Rains) * P(It Rains) + P(Jamie goes to the gym | It Does Not Rain) * P(It Does Not Rain)
P(Jamie goes to the gym | It Rains) = 0.20 (given)
P(Jamie goes to the gym | It Does Not Rain) = 0.60 (given)
P(It Rains) = 0.40 (given)
P(It Does Not Rain) = 1 - P(It Rains) = 1 - 0.40 = 0.60
Substituting the values:
P(Jamie goes to the gym) = (0.20 * 0.40) + (0.60 * 0.60) = 0.08 + 0.36 = 0.44
Therefore, the probability that Jamie will go to the gym is 0.44 or 44%.
b. To find the probability that it rained given that Jamie goes to the gym, we can use Bayes' theorem:
P(It Rained | Jamie goes to the gym) = (P(Jamie goes to the gym | It Rains) * P(It Rains)) / P(Jamie goes to the gym)
We already know:
P(Jamie goes to the gym | It Rains) = 0.20 (given)
P(Jamie goes to the gym) = 0.44 (calculated in part a)
Substituting these values:
P(It Rained | Jamie goes to the gym) = (0.20 * 0.40) / 0.44 = 0.08 / 0.44 = 0.1818
Therefore, the probability that it rained given that Jamie goes to the gym is approximately 0.1818 or 18.18%.
c. To find the probability that it rained given that Jamie didn't go to the gym, we can again use Bayes' theorem:
P(It Rained | Jamie didn't go to the gym) = (P(Jamie didn't go to the gym | It Rains) * P(It Rains)) / P(Jamie didn't go to the gym)
We already know:
P(Jamie didn't go to the gym | It Rains) = 1 - P(Jamie goes to the gym | It Rains) = 1 - 0.20 = 0.80 (complementary probability)
P(Jamie didn't go to the gym) = 1 - P(Jamie goes to the gym) = 1 - 0.44 = 0.56 (complementary probability)
Substituting these values:
P(It Rained | Jamie didn't go to the gym) = (0.80 * 0.40) / 0.56 = 0.32 / 0.56 ≈ 0.5714
Therefore, the probability that it rained given that Jamie didn't go to the gym is approximately 0.5714 or 57.14%.