A box contains four red marbles, seven white marbles, and five blue marbles. If two marbles are drawn one at a time, find the probability that both marbles are white if the draws are made as follows:
(a) With replacement ____________ (b) Without replacement ______________
There is a total of 16 marbles and 7 are white.
With replacement: (7/16)*(7/16) = 49/256 = 0.1914
Without replacement: (7/16)*(6/15) = 42/240 = 7/40 = 0.1750
To find the probability, we need to calculate the ratio of successful outcomes to total outcomes.
(a) With replacement:
With replacement means that after each draw, we put the marble back into the box before making the next draw.
Total number of marbles = 4 (red) + 7 (white) + 5 (blue) = 16
The probability of drawing a white marble in the first draw = 7/16
The probability of drawing a white marble in the second draw = 7/16
Since the draws are independent (with replacement), we can multiply the probabilities of the individual draws:
P(both marbles are white with replacement) = (7/16) * (7/16) = 49/256
Therefore, the probability that both marbles are white with replacement is 49/256.
(b) Without replacement:
Without replacement means that after each draw, we do not put the marble back into the box before making the next draw.
Total number of marbles = 4 (red) + 7 (white) + 5 (blue) = 16
The probability of drawing a white marble in the first draw = 7/16
After the first draw, there are 15 marbles left in the box, including 6 white marbles.
The probability of drawing a white marble in the second draw, given that the first marble was white = 6/15
So, P(both marbles are white without replacement) = (7/16) * (6/15) = 42/240
Therefore, the probability that both marbles are white without replacement is 42/240, which can also be simplified to 7/40.