# Students life

### The problem

Given probability of students availability at lesson is 0.8
The number of students in group is 100
Find probability of availability for the next ranges:

1. More then 75 but less then 90
2. Not less then 75
3. No more then 74

### Solution

There is single solution that available for calculation of probability with range of parameters - Moivre–Laplace theorem. Integral equation is looks like:

$$\\ P\left \{ k_{1} \leqslant k \leqslant k_{2} \right \} \approx \Phi(x_{2}) - \Phi(x_{1}) \\ \\ x_{1} = \frac{k_{1} - np}{\sqrt{npq}} \; \; ; \; \; x_{2} = \frac{k_{2} - np}{\sqrt{npq}} \\ \\ \Phi(x) = \frac{1}{\sqrt{2\pi}}\int_{0}^{x}e^{-\frac{t^{2}}{2}} dt$$

For all subtasks there are next common values:

1. p = 0.8
2. q = 0.2
3. ## n = 100

4. k1 = 75
5. k2 = 90

So solution will looks like:

## $$\\ x_{1} = \frac{75-100\cdot 0.8}{\sqrt{100\cdot 0.8\cdot 0.2}} = -1.25 \; \; ; \; \; x_{2} = \frac{100-100\cdot 0.8}{\sqrt{100\cdot 0.8\cdot 0.2}} = 5 \\ \\ P\left \{ 75 \leqslant k \leqslant 100 \right \} \approx \Phi (5) - \Phi(-1.25) = \\ = \Phi (5) + \Phi(1.25) = 0.49999 + 0.39435 = 0.89$$
$$\\ x_{1} = \frac{74-100\cdot 0.8}{\sqrt{100\cdot 0.8\cdot 0.2}} = -1.5 \; \; ; \; \; x_{2} = \frac{0-100\cdot 0.8}{\sqrt{100\cdot 0.8\cdot 0.2}} = -20 \\ \\ P\left \{ 0 \leqslant k \leqslant 74 \right \} \approx - \Phi (1.5) + \Phi(20) = \\ = \Phi (5) + \Phi(1.25) = -0.43319 + 0.5 = 0.06681$$