an urn contains 40 red chips and sixty white chips. six chips are drawn out and discarded, and a seventh is drawn. what is the probability that the seventh chip is red?
To find the probability that the seventh chip drawn is red, we first need to determine the total number of chips remaining in the urn after the previous six chips were drawn and discarded.
The number of red chips remaining in the urn after six chips were drawn and discarded would be 40 minus the number of red chips that were already drawn. Similarly, the number of white chips remaining in the urn would be 60 minus the number of white chips that were already drawn.
Since we are interested in the probability of drawing a red chip on the seventh draw, we need to consider two cases:
Case 1: The seventh chip drawn is red.
In this case, we need to have at least one red chip remaining in the urn. The number of ways to choose at least one red chip from the remaining chips is (40 - x) choose 1, where x is the number of red chips already drawn.
The number of ways to choose any chip (red or white) from the remaining chips is (100 - x) choose 1.
So, the probability of drawing a red chip on the seventh draw in this case would be:
P1 = [(40 - x) choose 1] / [(100 - x) choose 1]
Case 2: The seventh chip drawn is white.
In this case, we need to have no red chips remaining in the urn. The number of ways to choose zero red chips from the remaining chips is (40 - x) choose 0 (which is equal to 1), and the number of ways to choose any chip (red or white) from the remaining chips is (100 - x) choose 1.
So, the probability of drawing a white chip on the seventh draw in this case would be:
P2 = [(40 - x) choose 0] / [(100 - x) choose 1] = 1 / [(100 - x) choose 1]
Finally, the probability of drawing a red chip on the seventh draw can be calculated as the sum of the probabilities from both cases:
P(red on seventh draw) = P1 + P2
Note that in this calculation, we need to know how many red chips were already drawn in the first six draws (represented by x), as it affects the probabilities in both cases.