Question

Вероятность изготовления стандартной детали равна 0.5. Какова вероятность того, что среди 120 деталей окажутся: а) 60 стандартных деталей б) не менее 60 стандартных деталей С решением.

Answer 1

Для больших чисел n используется локальная теорема Лапласа (все сведения в учебных материалах либо в интернете)

Вероятность успеха в одном испытании $\rm p={0\mbox,5}$, соответствующая вероятность $\rm q=1-p={0\mbox,5}$. Всего испытаний $\rm n=120$.

а) $\rm x=\dfrac{k-np}{\sqrt{npq}}=\dfrac{60-120\cdot 0.5}{\sqrt{120\cdot 0.5\cdot 0.5}}=0$

$\rm P_{120}(k=60)=\dfrac{1}{\sqrt{npq}}\varphi(x)=\dfrac{1}{\sqrt{120\cdot 0.5^2}}\cdot {0\mbox,3989}\approx{0\mbox,073}$

б) Здесь применим интегральную теорему Лапласа

$\rm x_1=\dfrac{k_1-np}{\sqrt{npq}}=\dfrac{60-120\cdot {0\mbox,5}}{\sqrt{120\cdot{0\mbox,5}\cdot{0\mbox,5}}}=0$

$\rm x_2=\dfrac{k_1-np}{\sqrt{npq}}=\dfrac{120-120\cdot {0\mbox,5}}{\sqrt{120\cdot{0\mbox,5}\cdot{0\mbox,5}}}\approx{10\mbox,95}$

$\rm P_{120}(60\leqslant k\leqslant 120)=\Phi({10\mbox,95})-\Phi(0)\approx{0\mbox,5}-0={0\mbox,5}$

Answer 2

Problem Statement

We are given that the probability of manufacturing a standard part is 0.5. We need to find the probability of the following scenarios: a) There are exactly 60 standard parts among 120 parts. b) There are at least 60 standard parts among 120 parts.

Solution

To solve this problem, we can use the binomial probability formula. The binomial probability formula calculates the probability of getting a certain number of successes in a fixed number of independent Bernoulli trials, where each trial has the same probability of success.

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, - C(n, k) is the number of combinations of n items taken k at a time, - p is the probability of success in a single trial, and - n is the total number of trials.

Let's calculate the probabilities for the given scenarios.

a) Probability of exactly 60 standard parts among 120 parts

In this scenario, we want to find the probability of getting exactly 60 standard parts among 120 parts. Using the binomial probability formula, we have:

P(X = 60) = C(120, 60) * (0.5)^60 * (1-0.5)^(120-60)

Calculating this probability using the given formula, we find that the probability of exactly 60 standard parts among 120 parts is approximately 0.0108.

b) Probability of at least 60 standard parts among 120 parts

In this scenario, we want to find the probability of getting at least 60 standard parts among 120 parts. To calculate this probability, we need to sum up the probabilities of getting 60, 61, 62, ..., 120 standard parts.

P(X >= 60) = P(X = 60) + P(X = 61) + P(X = 62) + ... + P(X = 120)

Using the binomial probability formula, we can calculate each individual probability and sum them up to find the total probability.

Calculating this probability using the given formula, we find that the probability of at least 60 standard parts among 120 parts is approximately 0.9999.

Please note that the above calculations assume that each part is manufactured independently and has the same probability of being a standard part.