Question

question 3
a bag contains 5 green marbles, 4 blue marbles and 1 red marble for a total of 10 marbles. three marbles are randomly drawn from the bag one at a time.
a. if the marbles are replaced after each random drawing, what is the probability that all three marbles drawn in sequence are all green?
b. if the marbles are not replaced after each random drawing, what is the probability that all three marbles drawn in sequence are all green?
c. explain why there might be a difference between the two probabilities determined in part (a) and part (b).

Explanation:

Step1: Calculate probability for replacement case

The probability of drawing a green marble in one draw when replacement occurs is the number of green marbles divided by the total number of marbles. There are 5 green marbles and 10 total marbles, so the probability of drawing a green marble in one draw is $P_1=\frac{5}{10}=\frac{1}{2}$. Since the draws are independent when replacement occurs, the probability that all three marbles are green is the product of the probabilities of drawing a green marble in each draw. So $P_{a}=(\frac{5}{10})\times(\frac{5}{10})\times(\frac{5}{10})=\frac{5^3}{10^3}=\frac{125}{1000} = 0.125$.

Step2: Calculate probability for non - replacement case

For the first draw, the probability of drawing a green marble is $\frac{5}{10}$. For the second draw, since one green marble has been removed and not replaced, there are 4 green marbles left and 9 total marbles, so the probability is $\frac{4}{9}$. For the third draw, there are 3 green marbles left and 8 total marbles, so the probability is $\frac{3}{8}$. The probability that all three marbles are green when there is no replacement is $P_{b}=\frac{5}{10}\times\frac{4}{9}\times\frac{3}{8}=\frac{5\times4\times3}{10\times9\times8}=\frac{60}{720}=\frac{1}{12}\approx0.083$.

Step3: Explain the difference

In the replacement case, the probability of drawing a green marble remains the same for each draw because the composition of the marbles in the bag does not change. In the non - replacement case, the probability of drawing a green marble changes for each subsequent draw as the number of green marbles and total marbles in the bag decreases. This change in probabilities for each draw in the non - replacement case leads to a lower overall probability of getting three green marbles compared to the replacement case.

Answer:

a. $0.125$
b. $\frac{1}{12}\approx0.083$
c. In replacement, draw probabilities are constant; in non - replacement, they change due to marble removal.