Given: A bag contains some white and some black balls, all combinations of balls being equally likely. The total number of balls in the bag is 10. If three balls are drawn at random without replacement and all of them are found to be black, the probability that the bag contains 1 white and 9 black balls is to be found.

Let's try to solve the problem step-by-step.

Step 1: Finding the total number of ways to draw 3 balls from a bag of 10 balls

The total number of ways to draw 3 balls from a bag of 10 balls can be found using the combination formula as follows:

nCr = n! / r! (n-r)!

where n = 10 (total number of balls in the bag) and r = 3 (number of balls to be drawn)

So, the total number of ways to draw 3 balls from a bag of 10 balls is:

10C3 = 10! / 3! (10-3)! = 120

Step 2: Finding the number of ways to draw 3 black balls from a bag containing 9 black balls

Since all the balls are equally likely to be drawn, the probability of drawing a black ball is 9/10 (as there are 9 black balls out of 10). So, the number of ways to draw 3 black balls from a bag containing 9 black balls can be found using the following formula:

nCr = n! / r! (n-r)!

where n = 9 (number of black balls in the bag) and r = 3 (number of black balls to be drawn)

So, the number of ways to draw 3 black balls from a bag containing 9 black balls is:

9C3 = 9! / 3! (9-3)! = 84

Step 3: Finding the number of ways to draw 3 black balls from a bag containing 8 black balls and 2 white balls

Since all the balls are equally likely to be drawn, the probability of drawing a black ball is 8/10 (as there are 8 black balls out of 10) and the probability of drawing a white ball is 2/10 (as there are 2 white balls out of 10). So, the number of ways to draw 3 black balls from a bag containing 8 black balls and 2 white balls can be found using the following formula:

nCr = n! / r! (n-r)!

where n = 8 (number of black balls in the bag), r = 3 (number of black balls to be drawn)

So, the number of ways to draw 3 black balls from a bag containing 8 black balls and 2 white balls is:

8C3 = 8! / 3! (8-3)! = 56

Step 4: Finding the probability that the bag contains 1 white and 9 black balls, given that 3 black balls are drawn

Let P(A) be the probability that the bag contains 1 white and 9 black balls, and P(B) be the probability that 3 black balls are drawn. Then, the probability that the bag contains 1 white and 9 black balls, given that 3 black balls are drawn, can be found using Bayes' theorem as follows:

P(A|B)