Question

ПОЖАЛУЙСТА ПОМОГИТЕ РЕШИТЬ ЗАДАЧУ ПО ТЕОРИИ ВЕРОЯТНОСТИ Ответ указать в 0.000 В первой урне 5 белых и 7 черных шаров, во второй урне 9 белых и 4 черных шаров, в третьей урне 5 белых и 10 черных шаров. Из первой и второй урн наудачу выбираются по одному шару и помещаются в третью урну. После этого из третьей урны извлекается один шар. Какова вероятность, что извлеченный из третьей урны шар является черным?

Answer 1

Возможно несколько ситуаций вытащить шар черного цвета из урн, чтобы из третьей вытащить потом черный 1)черный-черный-черный. 2) черный-белый-черный 3) белый-черный-черный 4)белый-белый-черный. 1 ситуация. Черный из первой урны 7/12=0,583 Черный из второй урны 4/13=0,308. Тогда из третьей- 12/17=0,706 Итого:0,583*0,308*0,706=0,127. 2 ситуация. Черный из первой-7/12=0,583. Белый из второй- 9/13=0,692 Черный из третьей 11/17=0,647. Итого: 0,583*0,692*0,647=0,261. 3 ситуация. Белый из первой урны- 5/12=0,417 Черный из второй-4/13=0,308 черный из третей урны- 11/17=0,647 Итого: 0,417*0,308*0,647=0,083. 4 ситуация. Белый из первой урны- 5/12=0,417. Белый из второй урны- 9/13=0,692 Черный из третьей- 10/17=0,588. Итого: 0,417*0,692*0,588=0,17. Находим сумму этих вероятностей, что и будет искомой вероятностью: 0,127+0,261+0,083+0,17=0,641. Ответ:0,641.

Answer 2

Problem Analysis

In this problem, we have three urns: the first urn contains 5 white and 7 black balls, the second urn contains 9 white and 4 black balls, and the third urn contains 5 white and 10 black balls. One ball is randomly chosen from both the first and second urns and placed into the third urn. Finally, one ball is randomly chosen from the third urn. We need to find the probability that the ball chosen from the third urn is black.

Solution

To solve this problem, we can use the concept of conditional probability. Let's break down the problem step by step:

1. Calculate the probability of choosing a black ball from the first urn. 2. Calculate the probability of choosing a black ball from the second urn. 3. Calculate the probability of choosing a black ball from the third urn, given that one black ball was added from the first urn and one black ball was added from the second urn.

Let's calculate each of these probabilities.

Probability of choosing a black ball from the first urn

The first urn contains 5 white and 7 black balls. The probability of choosing a black ball from the first urn is given by:

P(Black ball from first urn) = Number of black balls in the first urn / Total number of balls in the first urn

P(Black ball from first urn) = 7 / (5 + 7)

Probability of choosing a black ball from the second urn

The second urn contains 9 white and 4 black balls. The probability of choosing a black ball from the second urn is given by:

P(Black ball from second urn) = Number of black balls in the second urn / Total number of balls in the second urn

P(Black ball from second urn) = 4 / (9 + 4)

Probability of choosing a black ball from the third urn, given the balls added from the first and second urns

After one black ball is added from the first urn and one black ball is added from the second urn, the third urn contains 7 + 4 + 10 = 21 balls, out of which 2 are black. The probability of choosing a black ball from the third urn, given the balls added from the first and second urns, is given by:

P(Black ball from third urn | balls added from first and second urns) = Number of black balls in the third urn / Total number of balls in the third urn

P(Black ball from third urn | balls added from first and second urns) = 2 / 21

Probability of choosing a black ball from the third urn

Finally, we need to calculate the probability of choosing a black ball from the third urn. This can be done using the law of total probability, considering the probabilities of choosing balls from the first and second urns:

P(Black ball from third urn) = P(Black ball from third urn | balls added from first and second urns) * P(Black ball from first urn) + P(Black ball from third urn | balls added from first and second urns) * P(Black ball from second urn)

P(Black ball from third urn) = (2 / 21) * (7 / (5 + 7)) + (2 / 21) * (4 / (9 + 4))

Now we can calculate this expression to find the probability of choosing a black ball from the third urn.

Calculation

Let's calculate the probability using the given values:

P(Black ball from third urn) = (2 / 21) * (7 / 12) + (2 / 21) * (4 / 13)

P(Black ball from third urn) ≈ 0.0952

Therefore, the probability that the ball chosen from the third urn is black is approximately 0.0952.

Answer

The probability that the ball chosen from the third urn is black is approximately 0.0952.