A box contains 3 red marbles and 7 white marbles.A marble is drawn from the box,and a marble of the other color is then put into the box.A second marble is drawn from the box.Find the probability that the second marble is red.if both marbles were of same color.what is the probability that they are both white?
To solve this problem, we can break it down into two separate scenarios: when the first marble drawn is red and when the first marble drawn is white.
Scenario 1: When the first marble drawn is red.
In this case, we have 3 red marbles and 7 white marbles in the box. After drawing a red marble, we put a white marble into the box, so we still have 3 red marbles but now 8 white marbles. The probability of drawing a red marble first is 3/10.
Scenario 2: When the first marble drawn is white.
In this case, we have 3 red marbles and 7 white marbles in the box. After drawing a white marble, we put a red marble into the box, so now we have 4 red marbles and 7 white marbles. The probability of drawing a white marble first is 7/10.
The probability that the second marble is red if both marbles were of the same color can be calculated by finding the probability of drawing two red marbles and adding it to the probability of drawing two white marbles.
Calculating the probability of drawing two red marbles:
P(2 red) = P(red first) * P(red second, given red first) = (3/10) * (4/9) = 2/15
Calculating the probability of drawing two white marbles:
P(2 white) = P(white first) * P(white second, given white first) = (7/10) * (6/9) = 14/45
Therefore, the probability that the second marble is red if both marbles were of the same color is 2/15, and the probability that they are both white is 14/45.