Task 4. Among the rye seeds, there is $0.4\%$ weed seeds. What is the probability of finding 5 weed seeds when randomly selecting 5000 seeds?
Solution. Using the Poisson formula (23), we find for $\lambda=n p=20$.

$$
p_{5000}(5)=\frac{20^{5} \cdot e^{-20}}{5!}=0.000055
$$

Answer. 0.000055.

## Exercises

1. The probability that a phone call will be interrupted due to technical reasons is 0.005. What is the probability that out of 1000 phone calls, exactly 4 will be interrupted due to technical reasons?
2. The probability of a defective part being produced is 0.008. Find the probabilities that among 1000 produced parts there will be: a) 8 defective; b) no more than 2 defective.
3. An electronic device contains 500 components, each of which reliably operates for a year with a probability of 0.994. Find the probability that exactly 2 components will fail within a year, if they operate independently of each other.
4. The average number of calls received by a telephone exchange in one minute is 2. Find the probabilities that in 5 minutes the telephone exchange will receive: a) 2 calls; b) no less than 2 calls; c) less than 2 calls.
5. On average, 2 taxis arrive at a taxi stand in 15 minutes. Find the probabilities that in 30 minutes: a) 3 taxis will arrive; b) no more than 3 taxis will arrive; c) no taxis will arrive.

## Answers

1. 0.1753. 2. a) 0.1828; b) 0.0139. 3. 0.223. 4. a) 0.00225; b) 0.9995; c) 0.0005. 5. a) 0.1952; b) 0.4318; c) 0.0183.

## § 9. Laplace's Formula

$1^{\circ}$. The numerical implementation of the Bernoulli and Poisson formulas for large $\lambda$ is difficult. This gap is filled by Laplace's formula.

Theorem 9. If in each of the $n$ independent trials the probability of event $A$ is the same value $p(0<p<1)$, then the probability $p_{n}(k)$ that event $A$ will occur exactly $k$ times is approximately expressed by the formula

$$
p_{n}(k)=\frac{\varphi(x)}{\sqrt{n p q}}
$$

where $x=\frac{k-n p}{\sqrt{n p q}}$, and $\varphi(x)=\frac{1}{\sqrt{2 \pi}} e^{-x^{2} / 2}$.

Laplace's formula for $p_{n}(k)$ is associated with the calculation of the Laplace function $\varphi(x)$ for certain values of $x$. The function $\varphi(x)$ is tabulated, and its values for some $x$ are given in Table 2 of the appendix. It is also worth noting that $\varphi(x)$ is an even function, i.e., $\varphi(-x)=\varphi(x)$, and $\varphi(x) \approx 0$ for values of $x$ greater than 4.

$2^{\circ}$ The function $\Phi(x)=\int_{0}^{x} \varphi(t) d t=\frac{1}{\sqrt{2 \pi}} \int_{0}^{x} e^{-t^{2} / 2} d t$ is called the integral Laplace function, and $\varphi(x)$ is called the differential Laplace function.

The function $\Phi(x)$ is used to calculate the approximate value of the probability that event $A$ will occur from $k_{1}$ to $k_{2}$ times in $n$ independent trials, if the probability of the occurrence of $A$ in each trial is $p, 0<p<1$.

Theorem 10. The approximate equality holds

$$
p_{n}\left(k_{1} \leqslant k \leqslant k_{2}\right)=\Phi\left(x_{2}\right)-\Phi\left(x_{1}\right)
$$

where $x_{1}=\frac{k_{1}-n p}{\sqrt{n p q}}, x_{2}=\frac{k_{2}-n p}{\sqrt{n p q}}$, and the number $n$ is large.

The function $\Phi(x)$ is odd, i.e., $\Phi(-x)=-\Phi(x)$. Its values for some values of $x$ are given in Table 1 of the appendix.

For values of $x$ greater than 5, the values of $\Phi(x)$ are close to 0.5.

Theorems 9 and 10 are also known as the local and integral theorems of Moivre-Laplace, respectively.

## Problems with Solutions