There are two jars, A and B. Jar A contains 7 red and 2 blue marbles, while jar B contains 5 red and 4 blue marbles. You roll a fair die.If you get a 2 or a 3, you pick a random marble from jar A and put it in jar B. You then pick a random marble from jar B.
Else you pick a random marble from jar B and put it in jar A. You then pick a random marble from jar A.
a.What is the probability that you end up with a blue marble?
b.What is the probability that you moved a blue marble, given you ended with a blue marble?
c.What is the probability that you rolled a 2 or 3, given that you ended with a blue marble?
d.What is the probability that you end with a blue marble, given that you moved a red marble?
a. 7/2
To answer the given questions, we will need to calculate the probabilities step by step. Let's break down each question and explain how to solve it.
a. What is the probability that you end up with a blue marble?
To find the probability of ending up with a blue marble, we need to consider two scenarios and add up their probabilities:
1. Picking a blue marble from jar A and then a blue marble from jar B.
2. Picking a red marble from jar A and then a blue marble from jar A.
For the first scenario, we need to calculate the probability of picking a blue marble from jar A (2 blue marbles out of a total of 9 marbles) and then multiply it by the probability of picking a blue marble from jar B (4 blue marbles out of a total of 9 marbles).
P(Blue from A and then Blue from B) = (2/9) * (4/9)
For the second scenario, we need to calculate the probability of picking a red marble from jar A (7 red marbles out of a total of 9 marbles) and then multiply it by the probability of picking a blue marble from jar A (2 blue marbles out of a total of 10 marbles after adding the red marble).
P(Red from A and then Blue from A) = (7/9) * (2/10)
Finally, we add up the probabilities of both scenarios to get the overall probability of ending up with a blue marble:
P(Blue) = P(Blue from A and then Blue from B) + P(Red from A and then Blue from A)
b. What is the probability that you moved a blue marble, given you ended with a blue marble?
To answer this question, we need to find the probability of moving a blue marble given that we ended up with a blue marble. This can be calculated by dividing the probability of moving a blue marble and ending up with a blue marble by the probability of ending up with a blue marble, which is calculated in the previous question.
c. What is the probability that you rolled a 2 or 3, given that you ended with a blue marble?
To find the probability of rolling a 2 or 3 given that we ended with a blue marble, we need to calculate the conditional probability. We divide the probability of rolling a 2 or 3 and ending up with a blue marble by the probability of ending up with a blue marble, which is calculated in question a.
d. What is the probability that you end with a blue marble, given that you moved a red marble?
To find the probability of ending with a blue marble given that we moved a red marble, we need to calculate the conditional probability. We divide the probability of moving a red marble and ending with a blue marble by the probability of moving a red marble, which we can calculate by considering the two scenarios:
1. Picking a blue marble from jar A and then moving it to jar B.
2. Picking a red marble from jar B.
Similar to question a, we can calculate the probabilities for each scenario and add them up. Finally, we divide the probability of moving a red marble and ending with a blue marble by the probability of moving a red marble.