# Probability - Amount of money in pocket

#### mathmari

##### Well-known member
MHB Site Helper
Hey!!

The amount of money a student in the Accounting Department has in his pocket is a random variable that follows the normal distribution, with an average price of $30$ euros and a variance of $100$.

a) What is the probability that a student has $25$ to $35$ euros in his pocket?

b) If we randomly select $25$ students, then what is the probability that a student has less than $20$ euros in his pocket?

c) How much money does $75\%$ of students have in their pocket?

I have done the following :

a) We have that the standard deviation is equal to $\sqrt{100}=10$.

Do we have to use the $z$-score?

We have that $Z=\frac{X-\mu}{\sigma}=\frac{X-30}{10}$.
Then \begin{align*}P(25\leq X\leq 35)&=P(X\leq 35)-P(X\leq 25)\\ & =P\left (Z\leq \frac{35-30}{10}\right )-P\left (Z\leq \frac{25-30}{10}\right )\\ & =P\left (Z\leq 0.5\right )-P\left (Z\leq 0.5\right )\end{align*} Is that correct so far?

b) Could you give me a hint?