For N balls, m are black and the rest white,what is the probability that in the first n balls from N, 1 of them are black.
Here m< N.
To find the probability that exactly one of the first n balls drawn from N balls is black, we need to follow these steps:
1. Calculate the total number of ways to select n balls from N:
This can be done using the combination formula or the binomial coefficient. The formula is given by:
C(N, n) = N! / (n!(N-n)!)
2. Calculate the number of ways to select exactly one black ball and the remaining (n-1) white balls from m black balls and (N-m) white balls:
This can be done using the combination formula. The formula is given by:
C(m, 1) * C(N-m, n-1)
3. Calculate the probability by dividing the number of favorable outcomes (step 2) by the total number of possible outcomes (step 1):
P = (C(m, 1) * C(N-m, n-1)) / C(N, n)
Let's consider an example to understand this better:
Suppose we have 10 balls in total (N = 10), out of which 5 are black (m = 5).
We want to find the probability of drawing exactly one black ball in the first 3 draws (n = 3).
1. The total number of ways to select 3 balls from 10 is given by:
C(10, 3) = 10! / (3!(10-3)! = 120
2. The number of ways to select exactly one black ball (out of 5) and two white balls (out of 5):
C(5, 1) * C(10-5, 3-1) = 5 * 6 = 30
3. The probability is given by:
P = 30 / 120 = 0.25 or 25%
So, in this example, the probability of drawing exactly one black ball in the first 3 draws is 0.25 or 25%.