A bag contains 2 red balls, 3 blue balls and 5 green balls.
3 balls are drawn at random. Find the probability that
a) the 3 balls are of different colours
b) 2 balls are of the same colour
c) all the 3 are of the same colour.
a. Pr(three different)=2/10*3/9*5/8
b. Pr(two same)=pr(none same)-pr(threesame)
= Pr(3 different)-3/10*2/9-1/8 - 5/10*4/9*3/8
To find the probability of these events, we need to calculate the total number of possible outcomes and the number of favorable outcomes for each event.
a) To find the probability that the 3 balls are of different colors, we first need to determine the total number of outcomes. We can choose 3 balls out of 10 without regard to order, which can be calculated using the combination formula:
Total outcomes = (10 choose 3) = 10! / (3! * (10-3)!) = 120
Next, we need to determine the number of favorable outcomes, i.e., the number of ways to select 3 balls of different colors.
Number of ways to choose 1 red, 1 blue, and 1 green ball = (2 choose 1) * (3 choose 1) * (5 choose 1) = 2 * 3 * 5 = 30
Therefore, the probability of selecting 3 balls of different colors is:
Probability = favorable outcomes / total outcomes = 30 / 120 = 1/4 = 0.25
b) To find the probability that 2 balls are of the same color, we again need to calculate the total number of outcomes. Choosing 2 balls out of 10 can be done in (10 choose 2) ways:
Total outcomes = 10! / (2! * (10-2)!) = 45
Next, we need to determine the number of favorable outcomes, which is the sum of the number of ways to choose 2 red balls, 2 blue balls, or 2 green balls:
Number of ways to choose 2 red balls = (2 choose 2) = 1
Number of ways to choose 2 blue balls = (3 choose 2) = 3
Number of ways to choose 2 green balls = (5 choose 2) = 10
Total favorable outcomes = 1 + 3 + 10 = 14
Therefore, the probability of selecting 2 balls of the same color is:
Probability = favorable outcomes / total outcomes = 14 / 45 = 14/45 ≈ 0.3111
c) To find the probability that all 3 balls are of the same color, we need to calculate the number of favorable outcomes, which is the sum of the number of ways to choose 3 red balls, 3 blue balls, or 3 green balls:
Number of ways to choose 3 red balls = (2 choose 3) = 0
Number of ways to choose 3 blue balls = (3 choose 3) = 1
Number of ways to choose 3 green balls = (5 choose 3) = 10
Total favorable outcomes = 0 + 1 + 10 = 11
Therefore, the probability of selecting 3 balls of the same color is:
Probability = favorable outcomes / total outcomes = 11 / 120 ≈ 0.0917