Question

If you are randomly sampling one at a time with replacement for a bag that contains eight blue marbles, seven red marbles, and five green marbles, what is the probability of obtaining

a. A blue marble in one draw from the bag?

b. Three blue marbles in three draws from the bag?

c. A red, a green, and a blue marble in that order in three draws from the bag?

d. At least two red marbles in three draws from the bag?

Answer 1

The question involves calculating the probability of various outcomes when drawing marbles from a bag with replacement. The probabilities range from simple single draws to combinations and dependent events over multiple draws.

Step-by-step explanation:

The student's question revolves around the concept of probability with replacement in a bag of marbles of different colors. The task involves calculating the chances of different outcomes when marbles are drawn one at a time with the chance for each outcome being reset after every draw due to replacement.

1. Probability of drawing a blue marble in one draw: There are eight blue marbles out of a total of 20 marbles (8 blue + 7 red + 5 green), so the probability is 8/20 or 2/5.
2. The probability of drawing three blue marbles in three draws: Since each draw is independent due to replacement, the probability is (8/20) * (8/20) * (8/20) = 512/8000 or 64/1000.
3. Probability of drawing a red, a green, and a blue marble in that order: This is calculated by multiplying the probability of each individual event. The odds for red is 7/20, for green is 5/20, and for blue is 8/20, so the combined probability is (7/20) * (5/20) * (8/20) = 280/8000 or 7/200.
4. Probability of drawing at least two red marbles in three draws: This can occur in three scenarios - RRB, RBR, or BRR. The probability for RRB is (7/20) * (7/20) * (13/20), for RBR is (7/20) * (13/20) * (7/20), and for BRR is (13/20) * (7/20) * (7/20). Adding these probabilities gives 301/2000.