below is the problem I have translated into English from Slovenian:
Verjetnost okvare tovornjaka je 0.6. Z neenacbo Cebiševa in Laplaceovim izrekom cenite verjetnost, da se izmed 5400 tovornjakov pokvari med 3136 in 3312 tovornjakov?
English: The probability that a truck will brake down is 0.6.
With inequality of Cebiš and Lapceov theorem determine the probability, that between 5400 trucks breaks down between 3136 and 3412 trucks ?
Please help !
To solve this problem, we need to use Chebyshev's inequality and Laplace's theorem to estimate the probability.
Let's first define our variables:
- p = probability that a truck breaks down = 0.6
- n = total number of trucks = 5400
- x = number of trucks that break down between 3136 and 3312
Chebyshev's inequality states that for any random variable with mean μ and standard deviation σ, the probability that the random variable deviates from the mean by k or more standard deviations is at most 1/k^2. We can use this inequality to estimate the probability that x falls within the given range.
To apply Chebyshev's inequality, we need to compute the mean and standard deviation of the number of trucks that break down.
Mean:
The mean of a binomial distribution is given by μ = n * p. In this case, μ = 5400 * 0.6 = 3240.
Standard Deviation:
The standard deviation of a binomial distribution is given by σ = sqrt(n * p * (1 - p)). In this case, σ = sqrt(5400 * 0.6 * 0.4) ≈ 47.4341649.
Now, let's compute the number of standard deviations x is from the mean:
z = (x - μ) / σ
In this case, since we are interested in the range between 3136 and 3312 trucks, we need to compute the value of z for both ends of the range:
z1 = (3136 - 3240) / 47.4341649 ≈ -2.1732715
z2 = (3312 - 3240) / 47.4341649 ≈ 1.526942
Now, let's use Laplace's theorem to estimate the probability that x falls within this range. According to Laplace's theorem, the probability can be estimated as the difference between the cumulative distribution function (CDF) at z2 and z1.
Since we are dealing with a binomial distribution, we can use the normal distribution as an approximation. We can look up the z-scores in a standard normal distribution table or use a calculator to find the corresponding probabilities. Subtracting the CDF values will give us the desired probability.
Now, let's calculate the probability using the cumulative distribution function (CDF) of the standard normal distribution.
P(3136 ≤ x ≤ 3312) = CDF(z2) - CDF(z1)
Using a standard normal distribution table or a calculator, we can find the corresponding probabilities:
CDF(z2) ≈ 0.93827
CDF(z1) ≈ 0.01505
P(3136 ≤ x ≤ 3312) ≈ 0.93827 - 0.01505 ≈ 0.92322
Therefore, the estimated probability that between 5400 trucks, between 3136 and 3412 trucks break down is approximately 0.92322.