Question

Сначала кубик бросается два раза. Если при этом хотя бы один раз выпадет 1 , то кубик бросается в третий раз. Найдите вероятность того, что 1 выпала ровно два раза. Ответ округлите до тысячных

Answer 1

Ответ:

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Объяснение:

Answer 2

Problem Analysis

We are given that a fair six-sided die is rolled twice. If at least one of the rolls results in a 1, the die is rolled a third time. We need to find the probability that a 1 is rolled exactly twice.

Solution

To find the probability of rolling a 1 exactly twice, we need to consider the possible outcomes of the three rolls. There are three cases to consider:

1. The first two rolls do not result in a 1, and the third roll is a 1. 2. The first roll is a 1, the second roll is not a 1, and the third roll is a 1. 3. The first roll is not a 1, the second roll is a 1, and the third roll is a 1.

Let's calculate the probability for each case and then sum them up to find the overall probability.

Case 1: The first two rolls do not result in a 1, and the third roll is a 1.

The probability of not rolling a 1 on a single roll is 5/6. Therefore, the probability of not rolling a 1 on the first two rolls is (5/6) * (5/6) = 25/36. The probability of rolling a 1 on the third roll is 1/6. Therefore, the probability of this case is (25/36) * (1/6) = 25/216.

Case 2: The first roll is a 1, the second roll is not a 1, and the third roll is a 1.

The probability of rolling a 1 on the first roll is 1/6. The probability of not rolling a 1 on the second roll is 5/6. The probability of rolling a 1 on the third roll is 1/6. Therefore, the probability of this case is (1/6) * (5/6) * (1/6) = 5/216.

Case 3: The first roll is not a 1, the second roll is a 1, and the third roll is a 1.

The probability of not rolling a 1 on the first roll is 5/6. The probability of rolling a 1 on the second roll is 1/6. The probability of rolling a 1 on the third roll is 1/6. Therefore, the probability of this case is (5/6) * (1/6) * (1/6) = 5/216.

Overall Probability

To find the overall probability, we sum up the probabilities of the three cases: (25/216) + (5/216) + (5/216) = 35/216.

Therefore, the probability of rolling a 1 exactly twice is 35/216.

Answer

The probability of rolling a 1 exactly twice when a fair six-sided die is rolled according to the given conditions is 35/216.