Question

В пирамиде находилось 5 ружей, два из которых имели оптические прицелы. Солдат взял наугад одно из ружей и попал в цель. какова вероятность что он стрелял из оптического ружья, если для него вероятность попадания из ружья с оптическим прицелом = 0,8, а без него = 0,5?

Answer 1

Одновременные события - солдат должен взять ружьё с оптикой (1 событие) и попасть в цель (2 событие).

Вероятность того, что взял с оптикой 2/5 = 0,4. Вероятность того, что попал 0,8.

Вероятность того, что попал из оптики 0,4*0,8 = 0,32.

Answer 2

Problem Analysis

In this problem, we are given a pyramid with 5 rifles, two of which have optical sights. A soldier randomly selects one rifle and hits the target. We need to find the probability that the soldier shot from a rifle with an optical sight, given that the probability of hitting the target with a rifle with an optical sight is 0.8, and without an optical sight is 0.5.

Solution

Let's denote the event of shooting from a rifle with an optical sight as A, and the event of hitting the target as B. We need to find the conditional probability P(A|B), which represents the probability of shooting from a rifle with an optical sight given that the target was hit.

To solve this problem, we can use Bayes' theorem, which states that:

P(A|B) = (P(B|A) * P(A)) / P(B)

Here, P(B|A) represents the probability of hitting the target given that the soldier shot from a rifle with an optical sight, P(A) represents the probability of shooting from a rifle with an optical sight, and P(B) represents the probability of hitting the target.

Given that P(B|A) = 0.8, P(A) = 2/5 (since there are 2 rifles with optical sights out of 5 rifles), and P(B) can be calculated using the law of total probability, we can substitute these values into Bayes' theorem to find P(A|B).

Calculation

To calculate P(B), we can use the law of total probability, which states that:

P(B) = P(B|A) * P(A) + P(B|A') * P(A')

Here, P(B|A') represents the probability of hitting the target given that the soldier shot from a rifle without an optical sight, and P(A') represents the probability of shooting from a rifle without an optical sight.

Given that P(B|A') = 0.5 and P(A') = 3/5 (since there are 3 rifles without optical sights out of 5 rifles), we can substitute these values into the law of total probability to find P(B).

Now, let's calculate P(B):

P(B) = P(B|A) * P(A) + P(B|A') * P(A')

P(B) = 0.8 * (2/5) + 0.5 * (3/5)

P(B) = 0.32 + 0.3

P(B) = 0.62

Now, let's substitute the values into Bayes' theorem to find P(A|B):

P(A|B) = (P(B|A) * P(A)) / P(B)

P(A|B) = (0.8 * (2/5)) / 0.62

P(A|B) = 0.32 / 0.62

P(A|B) ≈ 0.5161

Answer

The probability that the soldier shot from a rifle with an optical sight, given that the target was hit, is approximately 0.5161.