Question

3.1Вероятность малому предприятию быть банкротом равна 0,2. Найти вероятность того, что из восьми малых предприятий сохранятся более двух. Вероятность поломки (выхода из строя) в течение дня у каждого из трех работающих самосвалов равна соответственно р1, р2, р3. Какова вероятность того, что в течение рабочего дня:а) все самосвалы выйдут из строя;б) ни один не выйдет из строя;в) хотя бы один выйдет из строя;г) точно один выйдет из строя.p10,05p20,15p30,251.1В лотерее 10 билетов, из которых 4 выигрышных. Какова вероятность выиграть хотя бы один раз, купив 3 билета?

Answer 1

Решение ко второй задаче)

Answer 2

Probability of a Small Business Going Bankrupt

The probability of a small business going bankrupt is given as 0.2. To find the probability that more than two out of eight small businesses will survive, we can use the binomial distribution formula.

Let's denote the probability of a small business surviving as p and the probability of going bankrupt as q (1 - p). We can calculate the probability using the formula:

P(X > 2) = 1 - P(X <= 2)

where X follows a binomial distribution with parameters n (number of trials) and p (probability of success).

In this case, n = 8 and p = 0.2. Let's calculate the probability:

P(X <= 2) = C(8, 0) * (0.2)^0 * (0.8)^8 + C(8, 1) * (0.2)^1 * (0.8)^7 + C(8, 2) * (0.2)^2 * (0.8)^6

where C(n, r) represents the number of combinations of n items taken r at a time.

Using the formula, we can calculate the probability of more than two small businesses surviving.

Probability of All Dump Trucks Breaking Down

The probability of each dump truck breaking down during the day is given as p1, p2, and p3 for the three working dump trucks. To calculate the probability of different scenarios, we need to consider the individual probabilities and apply the appropriate calculations.

a) Probability that all dump trucks break down: To calculate the probability that all dump trucks break down, we multiply the individual probabilities of each truck breaking down:

P(all break down) = p1 * p2 * p3

b) Probability that none of the dump trucks break down: To calculate the probability that none of the dump trucks break down, we multiply the individual probabilities of each truck not breaking down:

P(none break down) = (1 - p1) * (1 - p2) * (1 - p3)

c) Probability that at least one dump truck breaks down: To calculate the probability that at least one dump truck breaks down, we can use the complement rule:

P(at least one breaks down) = 1 - P(none break down)

d) Probability that exactly one dump truck breaks down: To calculate the probability that exactly one dump truck breaks down, we need to consider all possible combinations of one truck breaking down while the others do not. We can calculate this using the binomial distribution formula:

P(exactly one breaks down) = C(3, 1) * p1 * (1 - p2) * (1 - p3) + C(3, 1) * (1 - p1) * p2 * (1 - p3) + C(3, 1) * (1 - p1) * (1 - p2) * p3

where C(n, r) represents the number of combinations of n items taken r at a time.

Now, let's calculate the probabilities based on the given values of p1, p2, and p3.

Probability of Winning in a Lottery

In a lottery, there are 10 tickets, out of which 4 are winning tickets. We want to calculate the probability of winning at least once by buying 3 tickets.

To calculate the probability of winning at least once, we can use the complement rule:

P(winning at least once) = 1 - P(not winning at all)

The probability of not winning with a single ticket is given by:

P(not winning with one ticket) = (6/10)

Since we are buying 3 tickets, the probability of not winning at all is:

P(not winning at all) = (6/10)^3

Finally, we can calculate the probability of winning at least once:

P(winning at least once) = 1 - (6/10)^3

Please note that the above calculations assume that each ticket has an equal chance of being chosen and that the draws are independent.

I hope this helps! Let me know if you have any further questions.