Question

В урне 20 шаров: 16 белых, 4 черных. Из урны вынимают сразу 3 шара. Какова вероятность того, что из них 2 шара будут белые и 1 черный.

Answer 1

Всего случаев: 20 * 19 * 18
Благоприятных случаев: 16 * 15 * 4
Вероятность: $\frac{16 * 15 * 4}{20 * 19 * 18} = \frac{8}{19 * 3} = \frac{8}{57}$

Answer 2

Problem Analysis

We have an urn with 20 balls, 16 of which are white and 4 are black. We are going to draw 3 balls from the urn. We need to find the probability that 2 of the balls drawn will be white and 1 will be black.

Solution

To find the probability, we need to calculate the number of favorable outcomes (drawing 2 white balls and 1 black ball) and divide it by the total number of possible outcomes (drawing any 3 balls from the urn).

Let's calculate the probability step by step:

Step 1: Calculate the number of ways to choose 2 white balls out of 16. - This can be calculated using the combination formula: C(n, k) = n! / (k! * (n-k)!), where n is the total number of balls and k is the number of balls we want to choose. - In this case, n = 16 (number of white balls) and k = 2. - So, the number of ways to choose 2 white balls out of 16 is C(16, 2) = 16! / (2! * (16-2)!) = 16! / (2! * 14!) = (16 * 15) / (2 * 1) = 120.

Step 2: Calculate the number of ways to choose 1 black ball out of 4. - Similarly, using the combination formula, the number of ways to choose 1 black ball out of 4 is C(4, 1) = 4! / (1! * (4-1)!) = 4! / (1! * 3!) = (4 * 3) / (1 * 1) = 12.

Step 3: Calculate the total number of ways to choose any 3 balls out of 20. - Again, using the combination formula, the number of ways to choose 3 balls out of 20 is C(20, 3) = 20! / (3! * (20-3)!) = 20! / (3! * 17!) = (20 * 19 * 18) / (3 * 2 * 1) = 1140.

Step 4: Calculate the probability. - The probability of drawing 2 white balls and 1 black ball is the ratio of the number of favorable outcomes to the total number of possible outcomes. - So, the probability is (number of ways to choose 2 white balls out of 16) * (number of ways to choose 1 black ball out of 4) / (number of ways to choose any 3 balls out of 20) = (120 * 12) / 1140 = 1440 / 1140 = 1.26.

Therefore, the probability that 2 of the balls drawn will be white and 1 will be black is approximately 1.26.

Answer

The probability that 2 of the balls drawn will be white and 1 will be black is approximately 1.26.