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

    For the first subtask:

  4. k1 = 75
  5. k2 = 90

So solution will looks like:

$$\\ x_{1} = \frac{75-100\cdot 0.8}{\sqrt{100\cdot 0.8\cdot 0.2}} = -1.25 \; \; ; \; \; x_{2} = \frac{90-100\cdot 0.8}{\sqrt{100\cdot 0.8\cdot 0.2}} = 2.5 \\ \\ P\left \{ 75 \leqslant k \leqslant 90 \right \} \approx \Phi (2.5) - \Phi(-1.25) = \\ = \Phi (2.5) + \Phi(1.25) = 0.49379 + 0.39435 = 0.89$$

For the second subtask:

  1. k1 = 75
  2. k2 = 100

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

For the third subtask:

  1. k1 = 0
  2. k2 = 74

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