Question

Suppose you have eight marbles in a bag - three that are green and five that are pink. Calculate the probabilities of each of the following events using the appropriate choice of either Equation (4.8) or Equation (4.10).

1. What is the probability that you draw a green marble, put it back, and then draw a pink marble?

2. What is the probability that you draw a green marble, do not put it back, and then draw a pink marble? [If you did not realize this already, somehow there has to be a difference between “replacement” versus “without replacement.”]

3. What is the probability that you draw two green marbles (without replacement)?

4. What is the probability that you draw two green marbles (without replacement) or you draw a green marble followed by a pink marble (without replacement)?

Answer 1

Given that there are 8 marbles in a bag. There are 3 green marbles and 5 pink marbles.

We know, Probability of any event = (Number of favorable outcomes)/(Total number of possible outcomes).

Now, we have to find the probability that we draw a green marble, put it back and then draw a pink marble. Thus, we have to find the probability of drawing a green marble and then a pink marble with replacement.

Now, the probability that we draw a green marble = 3/8[As, number of green marbles = 3 and total number of marbles = 8].

After replacing the green marble, the total number of marbles = 8. Now, number of pink marbles = 5. Thus, the probability that we draw a pink marble = 5/8.

Thus, the required probability = (3/8)*(5/8) = 15/64 = 0.2344(rounded up to four decimal places).

Thus, the probability we draw a green marble, put it back and then draw a pink marble = 0.2344 .

Now, we have to find the probability that we draw a green marble, do not put it back and then draw a pink marble. Thus, we have to find the probability of drawing a green marble and then pink marble without replacement.

Now, there are 8 marbles. Thus, the probability of drawing a green marble = 3/8. Now, as the marbles are selected without replacement, thus, after selecting the first marble, the total number of marbles = 7. Thus, the probability of drawing a pink marble from these 7 marbles = 5/7.

Thus, the required probability = (3/8)*(5/7) = 15/56 = 0.2679(rounded up to four decimal places).

Thus, the probability that we draw a green marble, do not put it back, and then draw a pink marble = 0.2679 . ---------(1)

Now, we have to find the probability that we draw two green marbles without replacement. Thus, the probability of drawing first green marble = 3/8. Now, after drawing the first marble, the number of marbles left = 7 and the number of green marbles left = 2. Thus, the probability of drawing second green marble = 2/7.

Thus, the required probability = (3/8)*(2/7) = 6/56 = 0.1071(rounded up to four decimal places).

Thus, the probability that we draw two green marbles without replacement = 0.1071 . -----(2)

Now, we have to find the probability that we draw two green marbles without replacement or we draw a green marble followed by a pink marble without replacement.

The probability that we draw two green marbles without replacement = 0.1071 [From (2)]

The probability that we draw a green marble followed by a pink marble without replacement = 0.2679 [From (1)].

Thus, the required probability = 0.1071 + 0.2679 = 0.375 .[As, P(A or B) = P(A) + P(B)]

Thus, the probability that we draw two green marbles without replacement or we draw a green marble followed by a pink marble without replacement = 0.375 .