# 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?

#### romsek

##### Member

b) given a set of independent Gaussian random variables what is the distribution of their sum? Don't know? Read your text.
Once you've established that the problem is the similar to (a)

c) Use the inverse of the CDF of the standard normal to find the z-score of 0.75. Convert the z-score into how much money using the mean and standard deviation as usual.

#### Prove It

##### Well-known member
MHB Math Helper
Are you allowed to use technology?

#### Country Boy

##### Well-known member
MHB Math Helper
I always considered paper and pencil "technology". They don't grow on trees!

#### Prove It

##### Well-known member
MHB Math Helper
I always considered paper and pencil "technology". They don't grow on trees!
Last time I checked, both paper and pencils do in fact come from trees...

But I digress, either the OP will need to use a calculator with a Normal Probability function on it to get the values, or else refer to a normal table. That is why I asked...

#### Country Boy

##### Well-known member
MHB Math Helper
Yes, "come from trees". Buy I said "grow on trees". I takes technology to convert trees to paper and pencils!

#### Country Boy

##### Well-known member
MHB Math 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 $35$ uueuros 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?
This seems to me to be ambiguous. Does it mean "at least one of the 25 students is less than 20 euros" or "all 25 have less than 20 Euros"?

c) How much money does $75\%$ of students have in their pocket?
Also ambiguous- the total of 75% of the students? And which 75%? It might be intended that 75% of the students have the same amount and that is what is being asked.

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?