Question

Вероятность независимых событий А и В равны соответственно 2/3 и 1/2. В результате испытания произошло ровно одно из них. Найдите вероятность того, что это было событие А.

Answer 1

Условная вероятность - формула Байеса

Р ( А | 1 ) = Р (1 | А ) * Р (А ) / Р (1)

Р ( 1 | А ) = Р ( не В ) = 1/2

Р (1) = Р ( А ) * Р ( не В ) + Р (В ) * Р ( не А ) = 2/3 * 1/2 + 1/2 * 1/3 = 1 / 2

Р ( А | 1 ) = 1/2 * 2/3 / ( 1/2 ) = 2/3

В этой задаче оказалось совпадение с вероятностью Р ( А ) . Это именно совпадение.

Найдем также вероятность

Р ( В | 1 ) чтобы убедится что сумма Р ( А | 1 ) + Р ( В | 1 ) = 1

Р ( В | 1 ) = Р (1 | В ) * Р ( В ) / Р ( 1 ) = Р ( не А ) * Р (В ) / Р ( 1 ) = 1/3 * 1/2 / ( 1/2 ) = 1/3

Проверка верна !

Answer 2

Probability of Independent Events A and B

To find the probability of event A given that exactly one of events A and B occurred, we need to use the concept of conditional probability.

Let's denote event A as A and event B as B. The probability of event A, P(A), is given as 2/3, and the probability of event B, P(B), is given as 1/2.

We are given that exactly one of events A and B occurred. This means that either event A occurred and event B did not, or event B occurred and event A did not.

To find the probability of event A given this information, we can use the formula for conditional probability:

P(A|B) = P(A ∩ B) / P(B)

Since the events A and B are independent, the probability of their intersection, P(A ∩ B), is equal to the product of their individual probabilities:

P(A ∩ B) = P(A) * P(B)

Substituting the given probabilities, we have:

P(A ∩ B) = (2/3) * (1/2) = 1/3

Now we can calculate the probability of event A given that exactly one of events A and B occurred:

P(A|B) = (1/3) / (1/2) = 2/3

Therefore, the probability that event A occurred given that exactly one of events A and B occurred is 2/3.