Question

Сим­мет­рич­ную мо­не­ту бро­са­ют 12 раз. Во сколь­ко раз ве­ро­ят­ность со­бы­тия «вы­па­дет ровно 4 орла» мень­ше ве­ро­ят­но­сти со­бы­тия «вы­па­дет ровно 5орлов»?​

Answer 1

Вот так вот, кажется:

Answer 2

Probability of getting exactly 4 heads

To find the probability of getting exactly 4 heads when flipping a symmetric coin 12 times, we need to calculate the number of ways this outcome can occur and divide it by the total number of possible outcomes.

The number of ways to get exactly 4 heads can be calculated using the binomial coefficient formula, which is given by:

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

Where: - n is the total number of trials (in this case, 12 flips) - k is the number of successful outcomes (in this case, 4 heads)

Using this formula, we can calculate the number of ways to get exactly 4 heads:

C(12, 4) = 12! / (4! * (12-4)!) = 12! / (4! * 8!) = (12 * 11 * 10 * 9) / (4 * 3 * 2 * 1) = 495

So, there are 495 ways to get exactly 4 heads when flipping a symmetric coin 12 times.

Probability of getting exactly 5 heads

Similarly, we can calculate the number of ways to get exactly 5 heads:

C(12, 5) = 12! / (5! * (12-5)!) = 12! / (5! * 7!) = (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1) = 792

So, there are 792 ways to get exactly 5 heads when flipping a symmetric coin 12 times.

Comparing the probabilities

To compare the probabilities, we need to divide the number of ways to get exactly 4 heads by the total number of possible outcomes and do the same for getting exactly 5 heads.

The total number of possible outcomes when flipping a symmetric coin 12 times is given by:

2^12 = 4096

Therefore, the probability of getting exactly 4 heads is:

P(4 heads) = 495 / 4096 ≈ 0.1206

And the probability of getting exactly 5 heads is:

P(5 heads) = 792 / 4096 ≈ 0.1934

So, the probability of getting exactly 4 heads is approximately 0.1206, while the probability of getting exactly 5 heads is approximately 0.1934.

Therefore, the probability of getting exactly 4 heads is less than the probability of getting exactly 5 heads.

Answer 3

Для решения этой задачи мы можем использовать формулу Бернулли, которая позволяет нам вычислить вероятность выпадения определенного количества успехов (в данном случае - орлов) при бросании монеты.

По формуле Бернулли вероятность выпадения k успехов из n испытаний при вероятности успеха p равна:

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

где C(n, k) - число сочетаний из n по k, p - вероятность успеха, (1-p) - вероятность неуспеха.

Для нашей задачи вероятность выпадения орла при бросании симметричной монеты равна 0.5.

Теперь мы можем рассчитать вероятность выпадения ровно 4 орлов при 12 бросках монеты:

P(4) = C(12, 4) * (0.5)^4 * (0.5)^(12-4) = 495 * 0.0625 * 0.0625 = 0.2441

А вероятность выпадения ровно 5 орлов при 12 бросках монеты:

P(5) = C(12, 5) * (0.5)^5 * (0.5)^(12-5) = 792 * 0.03125 * 0.03125 = 0.1934

Таким образом, вероятность события "выпадет ровно 4 орла" (0.2441) больше вероятности события "выпадет ровно 5 орлов" (0.1934) при 12 бросках монеты.