Question

Вероятность того, что батарейка нерабочая равна 0.05. Выбирают две таких батарейки. Какова вероятность, что обе они рабочие ? Почему нельзя сделать так: Найдём вероятность, что обе нерабочие 0,05^2 = 0.0025, а потом возьмём противоположное событие, то есть 1 - 0.0025 = 0.9975. Почему так нельзя ? Почему надо сразу же брать противоположное событие (1 - 0.05) и перемножать их ?

Answer 1

Потому что, ваша вероятность 1-0.0025 = 0.9975 это вероятность того, что хотя бы одна батарейка рабочая(т.е. рабочие оба батарейки или одна из них рабочая)

Вероятность того, что оба батарейки нерабочие, равна 0.05*0.05=0.0025, а вероятность того, что одна батарейка рабочая, а вторая не рабочая, равна 2*0,05*0,95=0,095

Вероятность того, что хотя бы одна батарейка нерабочая, равна 0.095+0.0025 = 0.0975

Вероятность того, что обе батарейки рабочие, равна 1-0.0975 = 0.9025

Второй способ быстрее. Вероятность того, что батарейка рабочая, равна 1-0,05 = 0,95. Тогда вероятность того, что обе батарейки рабочие, равна 0,95*0,95 = 0,9025

Answer 2

To calculate the probability that both batteries are working, you need to use the concept of independent events. When two events are independent, the probability of both events happening is the product of their individual probabilities.

Let's denote the event of a battery being working as "A" and the event of a battery being non-working (defective) as "Ā" (pronounced "A complement"). The probability of a battery being non-working is 0.05 (as given), so the probability of it being working is:

P(A) = 1 - P(Ā) = 1 - 0.05 = 0.95.

Now, since we are choosing two batteries and we want both of them to be working (event A), we calculate the probability of both events happening (working batteries) by multiplying their individual probabilities:

P(both batteries are working) = P(A) × P(A) = 0.95 × 0.95 ≈ 0.9025.

Now, let's address why the other approach you mentioned is incorrect. You suggested finding the probability of both batteries being non-working (0.05^2 = 0.0025) and then taking the complement to find the probability of both being working (1 - 0.0025 = 0.9975). This is not a valid approach because the events "both batteries are non-working" and "both batteries are working" are not mutually exclusive. They are complementary events, but they are not exhaustive.

In probability theory, complementary events are two events that cover all possible outcomes and do not overlap. For example, if we had only two possibilities: "working" and "non-working," then the probability of getting either a working battery or a non-working battery would be 1 (100%), and the events would be mutually exclusive.

However, in this case, there are more than two possible outcomes, such as one working and one non-working battery, or both working batteries. So, finding the probability of both batteries being non-working does not directly give you the probability of both being working. It only gives you the probability of one specific case out of several possibilities.

In summary, the correct approach is to find the individual probability of each event ("working" and "non-working") and then multiply those probabilities together to find the probability of both events happening simultaneously (both batteries are working).