Suppose an urn contains 8 red,5 white and 7 blue marbles.If 3 marbles are drawn at random from the urn with replacement,what is the probability that three marbles are the same color?Give answer in a reduced fraction.
could be RRR , WWW, or BBB
prob = (8/20)(7/19)(6/18) + (5/20)(4/19)(3/(18) + (7/20)(6/19)(5/18)
= 336/6840 + 60/6840 + 210/6840
= 606/6840
= 101/1140
To find the probability that three marbles of the same color are drawn from the urn with replacement, we need to calculate the probability for each individual color (red, white, and blue) and then add them together.
1. Probability of drawing three red marbles:
The probability of drawing one red marble is 8/20 (since there are 8 red marbles out of 20 total marbles), and there are three draws, so we need to multiply the probabilities: (8/20) * (8/20) * (8/20) = 512/8000
2. Probability of drawing three white marbles:
Similarly, the probability of drawing one white marble is 5/20, so the probability of drawing three white marbles is (5/20) * (5/20) * (5/20) = 125/8000
3. Probability of drawing three blue marbles:
The probability of drawing one blue marble is 7/20, so the probability of drawing three blue marbles is (7/20) * (7/20) * (7/20) = 343/8000
Now, we add up the probabilities for each individual color:
(512/8000) + (125/8000) + (343/8000) = 980/8000
Simplifying, the probability that three marbles of the same color are drawn is 7/56.
To find the probability that three marbles are the same color when drawn with replacement, we need to consider the probability of drawing three red marbles, three white marbles, or three blue marbles separately.
The probability of drawing a red marble is the ratio of the number of red marbles to the total number of marbles:
P(red) = (number of red marbles) / (total number of marbles) = 8 / (8 + 5 + 7) = 8 / 20 = 2/5
Since we're drawing with replacement, the event of drawing a red marble on each draw is independent. Therefore, the probability of drawing three red marbles is:
P(three red marbles) = P(red) * P(red) * P(red) = (2/5) * (2/5) * (2/5) = 8/125
Similarly, the probability of drawing three white marbles or three blue marbles can be computed as follows:
P(white) = (number of white marbles) / (total number of marbles) = 5 / (8 + 5 + 7) = 5 / 20 = 1/4
P(three white marbles) = P(white) * P(white) * P(white) = (1/4) * (1/4) * (1/4) = 1/64
P(blue) = (number of blue marbles) / (total number of marbles) = 7 / (8 + 5 + 7) = 7 / 20
P(three blue marbles) = P(blue) * P(blue) * P(blue) = (7/20) * (7/20) * (7/20) = 343/8000
Finally, to find the probability that three marbles are the same color, we sum up the probabilities of these three cases:
P(three marbles of the same color) = P(three red marbles) + P(three white marbles) + P(three blue marbles)
= 8/125 + 1/64 + 343/8000
= 64/1000 + 15/1000 + 343/8000
= 422/8000
The probability that three marbles drawn with replacement are of the same color is 422/8000.