A bag contains 9 red marbles, 4 blue marbles, and 7 yellow marbles. You randomly select three marbles from the bag. What is the probability that all three marbles are red when (a) you replace each marble before selecting the next marble, and (b) you do not replace each marble before selecting the next marble?
To find the probability of drawing three red marbles, we need to calculate the probability of each scenario.
(a) When you replace each marble before selecting the next marble:
In this case, the probability of drawing a red marble remains the same for each draw since the previous marble is placed back into the bag before the next draw.
To calculate the probability, we need to determine the probability of drawing one red marble, and multiply it by itself three times (because there are three draws with replacement).
Probability of drawing a red marble = Number of red marbles / Total number of marbles
Probability of drawing a red marble = 9 / (9 + 4 + 7)
Probability of drawing a red marble = 9 / 20
Probability of drawing three red marbles with replacement = (Probability of drawing a red marble) * (Probability of drawing a red marble) * (Probability of drawing a red marble)
Probability of drawing three red marbles with replacement = (9 / 20) * (9 / 20) * (9 / 20)
Probability of drawing three red marbles with replacement ≈ 0.151875
Therefore, the probability of drawing all three red marbles with replacement is approximately 0.151875 or 15.19%.
(b) When you do not replace each marble before selecting the next marble:
In this case, the probability of drawing a red marble changes for each draw since the previous marble is not returned to the bag before the next draw.
To calculate the probability, we need to determine the probability of drawing one red marble, and multiply it by the probability of drawing another red marble (with one red marble already drawn), and then multiply it by the probability of drawing the third red marble (with two red marbles already drawn).
Probability of drawing the first red marble = Number of red marbles / Total number of marbles
Probability of drawing the first red marble = 9 / (9 + 4 + 7)
Probability of drawing the first red marble = 9 / 20
Probability of drawing the second red marble = (Number of red marbles - 1) / (Total number of marbles - 1)
Probability of drawing the second red marble = (9 - 1) / (20 - 1)
Probability of drawing the second red marble = 8 / 19
Probability of drawing the third red marble = (Number of red marbles - 2) / (Total number of marbles - 2)
Probability of drawing the third red marble = (9 - 2) / (20 - 2)
Probability of drawing the third red marble = 7 / 18
Probability of drawing three red marbles without replacement = (Probability of drawing the first red marble) * (Probability of drawing the second red marble) * (Probability of drawing the third red marble)
Probability of drawing three red marbles without replacement = (9 / 20) * (8 / 19) * (7 / 18)
Probability of drawing three red marbles without replacement ≈ 0.134503
Therefore, the probability of drawing all three red marbles without replacement is approximately 0.134503 or 13.45%.
To calculate the probabilities, we need to know the total number of marbles in the bag for each case.
(a) When you replace each marble before selecting the next marble:
The total number of marbles in the bag is 9 red + 4 blue + 7 yellow = 20 marbles.
The probability of selecting a red marble is 9/20 for each selection because you are replacing the marble each time.
Therefore, the probability of selecting three red marbles is (9/20) * (9/20) * (9/20) = 0.10125, which is approximately 10.13%.
(b) When you do not replace each marble before selecting the next marble:
The total number of marbles decreases after each selection.
For the first selection, the probability of selecting a red marble is 9/20.
After selecting a red marble, there are 8 red marbles remaining out of 19 total marbles.
For the second selection, the probability of selecting a red marble is 8/19.
After selecting a red marble, there are 7 red marbles remaining out of 18 total marbles.
For the third selection, the probability of selecting a red marble is 7/18.
Therefore, the probability of selecting three red marbles without replacement is (9/20) * (8/19) * (7/18) = 0.13168, which is approximately 13.17%.
So, the probabilities are:
(a) 10.13% (approximately)
(b) 13.17% (approximately)
a) pr(3red, replacement)=(9/20)^3
b) pr(3red, wo replacement)=9/20 * 8/19*7/18