A box contains 10 red balls and 20 blue balls. Three balls are selected at random one after the other from the box, without replacement. If the first two balls selected are both blue, what is the probability that the third ball selected is red?
P(2 blue balls) = 20/30 * 19/29
P(red ball third) = 10/28
To find the probability that the third ball selected is red given that the first two balls selected are both blue, we can use conditional probability.
Step 1: Determine the total number of possible outcomes.
When selecting three balls from the box without replacement, the total number of possible outcomes is the number of ways to choose three balls out of 30 (10 red + 20 blue), which is given by the binomial coefficient (30 choose 3). This can be calculated using the formula:
C(n, r) = n! / (r! * (n-r)!)
where n is the total number of objects, and r is the number of objects we are selecting.
In this case, n = 30 and r = 3, so we have:
C(30, 3) = 30! / (3! * (30-3)!) = 30! / (3! * 27!) = (30 * 29 * 28) / (3 * 2 * 1) = 4060
So, there are 4060 possible outcomes when selecting three balls from the box.
Step 2: Determine the number of favorable outcomes.
Given that the first two balls selected are both blue, there are 20 blue balls remaining in the box, and 10 red balls. Selecting any one of these 10 red balls as the third ball would be considered a favorable outcome.
Therefore, the number of favorable outcomes is 10.
Step 3: Calculate the probability.
The probability of the third ball selected being red, given that the first two balls selected were both blue, is the ratio of the number of favorable outcomes to the total number of possible outcomes:
Probability = Number of favorable outcomes / Total number of possible outcomes
Plugging in the values we found, we have:
Probability = 10 / 4060 ≈ 0.0025
So, the probability that the third ball selected is red, given that the first two balls selected are both blue, is approximately 0.0025, or 0.25%.