An urn contains 7 red and 3 white marbles. Three marbles are drawn from the urn one after another.
Find the probability that first two are red and third is white if,
a) Marbles drawn are replaced after every draw.
b) Marbles drawn are not replaced after every draw.
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To find the probability in both cases, we need to consider the total number of possible outcomes and the favorable outcomes.
a) When marbles are replaced after every draw:
In this case, the probability of drawing a red marble on each draw is always going to be the same, since the urn is replenished with the same ratio of red and white marbles.
The probability of drawing a red marble on the first draw is 7/10.
The probability of drawing a red marble on the second draw is also 7/10.
The probability of drawing a white marble on the third draw is 3/10.
Since these events are independent, we can multiply the probabilities together to find the overall probability:
P(First two are red and third is white) = (7/10) * (7/10) * (3/10) = 0.147 or 14.7%
b) When marbles are not replaced after every draw:
With each draw, the total number of marbles decreases. Therefore, the probability of drawing a red marble on each draw changes.
The probability of drawing a red marble on the first draw is 7/10.
The probability of drawing a red marble on the second draw, without replacement, is 6/9.
The probability of drawing a white marble on the third draw, without replacement, is 3/8.
Again, since these events are independent, we can multiply the probabilities together to find the overall probability:
P(First two are red and third is white) = (7/10) * (6/9) * (3/8) = 0.14 or 14%
Therefore, the probability of drawing the first two marbles as red and the third marble as white is 14.7% when marbles are replaced after every draw, and 14% when marbles are not replaced after every draw.