Question

СРОЧНОО ДАЮ 40 БАЛЛОВ Під час тестування з математики учень мае розв'язати 5 задач. Імовірність туго, що він правильно розв'яже одну задачу, дорівнює 0,8. Щоб скласти тест, тучневі треба розв'язати не менше ніж 3 задачі. Знайдіть з точністю до 0,01 ймо- вірність того, що учень складе тест​

Answer 1

0.72528 или 72.53% (округлив до двух знаков после запятой).

Объяснение:

вероятность того, что он решит 3, 4 или 5 задач :

P(3 или более правильно решенных задач) = P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5)

Для вычисления вероятностей P(X = k), где k = 3, 4, 5, используем формулу биномиального распределения:

P(X = k) = C(n,k) * p^k * (1-p)^(n-k),

формуле - C(n,k) – количество комбинаций n элементов по k.

P(X = 3) = C(5,3) * 0.8^3 * (1-0.8)^(5-3) ≈ 0.288 P(X = 4) = C(5,4) * 0.8^4 * ( 1-0.8)^(5-4) ≈ 0.4096 P(X = 5) = C(5,5) * 0.8^5 * (1-0.8)^(5-5) ≈ 0.32768

P(3 или более правильно решенных задач) ≈ 0.288 + 0.4096 + 0.32768 ≈ 0.72528

вероятность того, что учащийся сдаст тест 0.72528 или 72.53%

Answer 2

Probability of Solving at Least 3 Problems

To find the probability that the student will pass the test by solving at least 3 problems correctly, we need to calculate the probability of solving exactly 3, 4, and 5 problems correctly and then sum them up.

The probability of solving exactly 3 problems correctly can be calculated using the binomial probability formula:

P(X=k) = C(n, k) * p^k * (1-p)^(n-k)

Where: - P(X=k) is the probability of getting exactly k successes - n is the total number of trials (5 in this case) - k is the number of successful trials (3 in this case) - p is the probability of success in a single trial (0.8 in this case) - C(n, k) is the number of combinations of n items taken k at a time

Using this formula, we can calculate the probability of solving exactly 3 problems correctly:

P(X=3) = C(5, 3) * 0.8^3 * (1-0.8)^(5-3)

Similarly, we can calculate the probabilities of solving exactly 4 and 5 problems correctly:

P(X=4) = C(5, 4) * 0.8^4 * (1-0.8)^(5-4)

P(X=5) = C(5, 5) * 0.8^5 * (1-0.8)^(5-5)

Calculating the Probabilities

Let's calculate these probabilities step by step:

1. Probability of solving exactly 3 problems correctly: - C(5, 3) = 5! / (3! * (5-3)!) = 10 - P(X=3) = 10 * 0.8^3 * (1-0.8)^(5-3) = 0.4096

2. Probability of solving exactly 4 problems correctly: - C(5, 4) = 5! / (4! * (5-4)!) = 5 - P(X=4) = 5 * 0.8^4 * (1-0.8)^(5-4) = 0.4096

3. Probability of solving exactly 5 problems correctly: - C(5, 5) = 5! / (5! * (5-5)!) = 1 - P(X=5) = 1 * 0.8^5 * (1-0.8)^(5-5) = 0.32768

Summing Up the Probabilities

To find the probability of passing the test by solving at least 3 problems correctly, we need to sum up the probabilities of solving exactly 3, 4, and 5 problems correctly:

P(X>=3) = P(X=3) + P(X=4) + P(X=5)

P(X>=3) = 0.4096 + 0.4096 + 0.32768

**P(X>=3) =