Question

Игральную кость бросают семь раз. Найти вероятность того, что «шестерка» появится ровно 4 раз. (использовать формулы теории вероятности)​

Answer 1

Ответ:

Объяснение:

При одном броске вероятность выпадения «шестёрки» равна  

р = 1/6, а  вероятность не выпадения равна 1-р = 5/6.  

Каждый бросок - независимое испытание.

Применим формулу Бернулли.

  где        n=7,  m=4

Answer 2

Ответ:

Объяснение:

Задача на схему Бернулли:

${\displaystyle P_{n}(m)=C_{n}^{m}p^{m}(1-p)^{n-m}}$

В нашем случае - n = 7, m = 4, p = 1/6, 1 - p = 5/6

${\displaystyle P_{7}(4)=C_{7}^{4}(\frac{1}{6})^{4}(\frac{5}{6})^{3}} = \frac{7*6*5*5^{3}}{1*2*3*6^{7}}$

≈ 0.015

Answer 3

Problem Statement

Игральную кость бросают семь раз. Найдем вероятность того, что «шестерка» появится ровно 4 раза.

Solution

To find the probability of getting a "six" exactly 4 times when rolling a die 7 times, we can use the binomial probability formula. The binomial probability formula is given by:

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 number of trials (in this case, the number of times the die is rolled) - k is the number of successes (in this case, the number of times "six" appears) - p is the probability of success on a single trial (in this case, the probability of getting a "six" on a single roll) - C(n, k) is the number of combinations of n items taken k at a time, also known as the binomial coefficient

In this case, n = 7 (the die is rolled 7 times), k = 4 (we want "six" to appear exactly 4 times), and p = 1/6 (the probability of getting a "six" on a single roll).

Let's calculate the probability using the formula:

P(X=4) = C(7, 4) * (1/6)^4 * (5/6)^(7-4)

Calculation

Using the formula, we can calculate the probability as follows:

P(X=4) = C(7, 4) * (1/6)^4 * (5/6)^3

To calculate the binomial coefficient C(7, 4), we can use the formula:

C(n, k) = n! / (k! * (n-k)!)

C(7, 4) = 7! / (4! * (7-4)!) = 7! / (4! * 3!) = (7 * 6 * 5) / (3 * 2 * 1) = 35

Substituting the values into the formula:

P(X=4) = 35 * (1/6)^4 * (5/6)^3

Calculating the probability:

P(X=4) = 35 * (1/6)^4 * (5/6)^3 ≈ 0.0694

Therefore, the probability of getting a "six" exactly 4 times when rolling a die 7 times is approximately 0.0694.