A box contains 2 white calls and 7 pink balls. Two balls are drawn at random from the box.
a) If the first ball is drawn and not replaced, what is the probability that two balls will be drawn of different colors?
b) If the first ball is drawn and then put back, what is the probability that two balls will be drawn of different colors?
P(WP) = (2/9)(7/8) = 7/36
P(PW) = (7/9)(2/*) = 7/36
Prob of your event = 7/36+7/36 = 7/18
if ball is replaced,
Prob(your event) = 2(2/9)(7/9) = 28/81
To find the probability of drawing two balls of different colors, we need to consider two scenarios: when the first ball is not replaced (in scenario a), and when the first ball is replaced (in scenario b).
a) If the first ball is drawn and not replaced:
In this case, there are a total of 9 balls in the box (2 white + 7 pink). When the first ball is drawn, the box is left with 8 balls, but the number of white and pink balls changes.
Since we want to find the probability of drawing two balls of different colors, we have two possibilities:
- Drawing a white ball first and then a pink ball
- Drawing a pink ball first and then a white ball
The probability of drawing a white ball first and then a pink ball can be calculated as follows:
Probability = (number of white balls / total number of balls) * (number of pink balls / total number of remaining balls)
Probability = (2/9) * (7/8)
Similarly, the probability of drawing a pink ball first and then a white ball is also (2/9) * (7/8).
Since these are the only two possibilities, we can add the probabilities together to get the final probability of drawing two balls of different colors:
Probability (two balls of different colors) = Probability (white first, pink second) + Probability (pink first, white second)
Probability (two balls of different colors) = 2 * (2/9) * (7/8)
b) If the first ball is drawn and then put back:
In this case, after drawing the first ball, it is put back in the box, so the total number of balls remains the same.
Again, we have two possibilities:
- Drawing a white ball first and then a pink ball
- Drawing a pink ball first and then a white ball
Since each draw is independent, the probability of each possibility remains the same as in scenario a:
Probability (white first, pink second) = (2/9) * (7/9)
Probability (pink first, white second) = (7/9) * (2/9)
Again, we can add these probabilities together to get the final probability of drawing two balls of different colors:
Probability (two balls of different colors) = Probability (white first, pink second) + Probability (pink first, white second)
Probability (two balls of different colors) = 2 * (2/9) * (7/9)
To summarize:
a) If the first ball is drawn and not replaced, the probability of drawing two balls of different colors is 2 * (2/9) * (7/8).
b) If the first ball is drawn and then put back, the probability of drawing two balls of different colors is 2 * (2/9) * (7/9).