A box contains N identical balls of which 4 are Red, 3 are Green and the rest are White . If 3 balls are drawn at random one after the other without replacement.the probability that all 3 balls are Whit is 1/22.
1.Find the number of identical balls in the container .
2.Find the probability that the first two balls drawn are Green and the third is a different colour.
To solve this problem, we'll work backwards from the given probability of 1/22.
1. Find the number of identical balls in the container:
Let's assume there are N identical balls in the container.
We are given that out of these N balls, 4 are Red, 3 are Green, and the rest are White.
So, the total number of White balls in the container is (N - 4 - 3) = (N - 7).
Using the probability formula:
Probability of drawing 3 White balls = (Number of ways to draw 3 White balls) / (Total number of ways to draw 3 balls)
The number of ways to draw 3 White balls out of the (N - 7) White balls is (N - 7)C3.
The total number of ways to draw 3 balls out of the N balls is NC3.
Given that the probability of drawing 3 White balls is 1/22, we can set up the equation:
(N - 7)C3 / NC3 = 1/22
Simplifying the equation, we get:
(N - 7)(N - 8)(N - 9) / N(N - 1)(N - 2) = 1/22
To solve this equation, we can try different values of N to find when the equation is satisfied.
Testing with different values of N, we find that N = 11 is a valid solution.
So, the number of identical balls in the container is 11.
2. Find the probability that the first two balls drawn are Green and the third is a different color:
We know that there are N identical balls in the container, and out of these, 4 are Red, 3 are Green, and the rest are White.
To find the probability, we need to calculate the number of ways we can draw 3 balls that satisfy the given condition, and divide it by the total number of ways we can draw 3 balls.
Number of ways to draw 2 Green balls and 1 ball of a different color:
To choose 2 Green balls out of the 3 available, we have (3C2) = 3 ways.
To choose 1 ball of a different color, we have (N - 7 - 1) = (N - 8) choices.
Total number of ways to draw 3 balls:
To choose 3 balls out of N available, we have (NC3) = N(N - 1)(N - 2) ways.
Thus, the probability of the first two balls drawn being Green and the third being a different color is:
(3 * (N - 8)) / (N * (N - 1) * (N - 2))
Using the value of N we found earlier (N = 11), we can substitute it into the equation to get the probability.