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 solve these questions, we can use probability rules and basic calculations. Let's calculate each question step by step:
a. What is the probability that Jamie will go to the gym?
We have two scenarios: when it rains and when it does not rain.
When it rains:
- Probability of rain: 40%
- Probability of going to the gym given rain: 20%
Therefore, the probability of Jamie going to the gym when it rains is 40% * 20% = 8%.
When it does not rain:
- Probability of no rain: 60%
- Probability of going to the gym given no rain: 60%
Therefore, the probability of Jamie going to the gym when it does not rain is 60% * 60% = 36%.
To calculate the overall probability of Jamie going to the gym, we sum the probabilities of each scenario: 8% + 36% = 44%.
So, the probability that Jamie will go to the gym is 44%.
b. If Jamie goes to the gym, what is the probability that it rained?
We can calculate this using Bayes' theorem.
Let's denote the following:
- R: Event that it rained
- G: Event that Jamie goes to the gym
We want to calculate P(R|G), which represents the probability that it rained given that Jamie went to the gym.
Using Bayes' theorem, we have:
P(R|G) = P(G|R) * P(R) / P(G)
P(G|R) represents the probability of Jamie going to the gym given that it rained, which is 20%; P(R) is the probability of rain, which is 40%; and P(G) is the overall probability of Jamie going to the gym, which we calculated as 44% in part a.
Therefore, P(R|G) = (20% * 40%) / 44% = 18.18%
So, if Jamie goes to the gym, the probability that it rained is approximately 18.18%.
c. If Jamie doesn't go to the gym, what is the probability that it rained?
Similarly, we will use Bayes' theorem to calculate P(R|not G), which represents the probability that it rained given that Jamie did not go to the gym.
P(R|not G) = P(not G|R) * P(R) / P(not G)
P(not G|R) represents the probability of Jamie not going to the gym given that it rained, which is 80% (100% minus the 20% probability she goes to the gym); P(R) is the probability of rain, which is 40%; and P(not G) is the overall probability of Jamie not going to the gym, which is the complement of P(G), i.e. 1 - 44% = 56%.
Therefore, P(R|not G) = (80% * 40%) / 56% = 57.14%
So, if Jamie doesn't go to the gym, the probability that it rained is approximately 57.14%.