I have a bag that contains 10 red marbles, 9 green marbles, and 6 blue marbles. I
reach into it and take one marble at random. Keeping that marble, I reach into the
bag again and take a second marble at random.
a. What is the probability that the first marble is green and the second one is blue?
b. What is the probability that the first marble is red given that the second one is red?
Hint: Use Bayes’ Theorem
a. P = 9/(10+9+6) = 9/25. green
P = 6/(10+8+6) = 6/24 = 1/4. blue.
To find the probability in both cases, we need to consider the total number of marbles and their colors.
a. To find the probability that the first marble is green and the second one is blue, we need to consider both events independently.
Step 1: Find the probability of drawing a green marble first.
There are a total of 10 red marbles, 9 green marbles, and 6 blue marbles, so the probability of drawing a green marble first is 9/25.
Step 2: Find the probability of drawing a blue marble second.
After taking out one marble, we are left with 24 marbles in the bag, consisting of 10 red marbles, 9 green marbles, and 5 blue marbles. So the probability of drawing a blue marble second is 5/24.
Step 3: Multiply the probabilities of both events.
Since the two events are independent, we can multiply their probabilities together:
P(green first and blue second) = P(green first) * P(blue second)
= (9/25) * (5/24)
= 45/600
= 3/40
Therefore, the probability that the first marble is green and the second one is blue is 3/40.
b. To find the probability that the first marble is red given that the second one is red, we need to use Bayes' Theorem.
Bayes' Theorem states:
P(A|B) = (P(B|A) * P(A)) / P(B)
In this case, let A be the event that the first marble is red, and B be the event that the second marble is red.
Step 1: Find P(B|A), the probability of the second marble being red given that the first marble is red.
Since the first marble is red, we have a total of 9 red marbles left out of 24 marbles. So the probability of drawing a red marble second, given that the first marble is red, is 9/24.
Step 2: Find P(A), the probability that the first marble is red.
Out of the original 25 marbles, 10 are red. So the probability of drawing a red marble first is 10/25.
Step 3: Find P(B), the probability that the second marble is red.
After removing one marble, we have 24 marbles left in the bag, including 9 red marbles. So the probability of drawing a red marble second is 9/24.
Step 4: Apply Bayes' Theorem to find P(A|B).
P(red first given that red second) = (P(red second given that red first) * P(red first)) / P(red second)
= (9/24) * (10/25) / (9/24)
= (9/24) * (10/25) / (9/24)
= 10/25
= 2/5
Therefore, the probability that the first marble is red given that the second one is red is 2/5.