A box contains 40 balls: 10 white, 25 black and 5 red balls. Three balls are extracted
from this box. What is the probability that the three balls are of different colors?
P(all white) = 10/40 * 9/39 * 8/38
P(all black) = 25/40 * 24/39 * 23/38
P(all red) = 5/40 * 4/39 * 3/38
P(all the same) is the sum of those three
P(not all one color) = 1 - P(all the same)
or
C(10,1) * C(25,1) * C(5,1) / C(40,3)
= 125/988
oobleck forgot to subtract all cases of something like RRB,
that is, 2 of one colour, 1 other, giving you cases of only 2 colours
To find the probability that the three balls are of different colors, we need to calculate the ratio of successful outcomes to the total number of possible outcomes.
Let's break it down step by step:
Step 1: Calculate the total number of possible outcomes.
There are a total of 40 balls in the box. So, for the first ball, you have 40 options to choose from. After one ball is chosen, there are 39 balls remaining for the second ball, and 38 balls remaining for the third ball. Therefore, the total number of possible outcomes is:
40 * 39 * 38 = 59,280
Step 2: Calculate the number of successful outcomes.
To have three balls of different colors, you need to select one ball of each color: white, black, and red.
For the first ball, you have 10 options among the white balls.
For the second ball, you have 25 options among the black balls.
For the third ball, you have 5 options among the red balls.
Therefore, the number of successful outcomes is:
10 * 25 * 5 = 1,250
Step 3: Find the probability.
To find the probability, divide the number of successful outcomes (1,250) by the total number of possible outcomes (59,280):
P(three balls of different colors) = 1,250 / 59,280 ≈ 0.02107
So, the probability that the three balls drawn are of different colors is approximately 0.02107, or 2.107%.