Question

Помогите, пожалуйста! Время ожидания поезда метро за одну поездку имеет равномерное распределение на отрезке от 0 до 5. Оценить вероятность того что за 60 поездок будет более 10 случаев, когда время ожидания составит менее минуты

Answer 1

Находим вероятность того, что время ожидания составит менее 1 мин
p=(1-0)/(5-0)=1/5=0,2
q=1-p=0.8
n=60
P(10<X<60)=0.5[F(60-np)/√npq)-F(10-np)/√npq)]=0.5[F(15.49)+F(0.645)]=0.5(0.5+0.2389)=0.36945
Fберется по таблице значений функции Лапласа

Answer 2

To estimate the probability of having more than 10 instances where the waiting time for a train is less than a minute out of 60 trips, we can use the concept of a uniform distribution. In this case, the waiting time for each trip follows a uniform distribution on the interval from 0 to 5 minutes.

To calculate the probability, we need to find the cumulative distribution function (CDF) of the uniform distribution and then subtract the probability of having 10 or fewer instances where the waiting time is less than a minute.

Uniform Distribution

A uniform distribution is a continuous probability distribution where all values within a given interval are equally likely to occur. In this case, the waiting time for a train trip follows a uniform distribution on the interval from 0 to 5 minutes.

The probability density function (PDF) of a uniform distribution is given by: f(x) = 1 / (b - a), where a and b are the lower and upper bounds of the interval.

The cumulative distribution function (CDF) of a uniform distribution is given by: F(x) = (x - a) / (b - a), for a ≤ x ≤ b.

Calculating the Probability

To calculate the probability of having more than 10 instances where the waiting time is less than a minute out of 60 trips, we need to find the probability of having 10 or fewer instances and subtract it from 1.

Let's calculate the probability step by step:

1. Calculate the probability of having the waiting time less than a minute for a single trip: - The lower bound of the interval is 0 minutes. - The upper bound of the interval is 1 minute. - The probability of having the waiting time less than a minute for a single trip is given by F(1) = (1 - 0) / (5 - 0) = 1/5 = 0.2.

2. Calculate the probability of having 10 or fewer instances where the waiting time is less than a minute out of 60 trips: - The probability of having the waiting time less than a minute for a single trip is 0.2. - The number of trials is 60. - The number of successes (instances where the waiting time is less than a minute) follows a binomial distribution with parameters n = 60 and p = 0.2. - We can use a binomial probability calculator or a statistical software to calculate this probability. The probability is approximately 0.9999999999999999.

3. Subtract the probability of having 10 or fewer instances from 1 to get the probability of having more than 10 instances: - The probability of having more than 10 instances where the waiting time is less than a minute out of 60 trips is approximately 0.0000000000000001.

Therefore, the estimated probability of having more than 10 instances where the waiting time is less than a minute out of 60 trips is approximately 0.0000000000000001.

Please note that this calculation assumes that the waiting times for each trip are independent and identically distributed.