8.2c Probability Of An Event

Example:

The masses of pears in a fruit stall are normally distributed with a mean of 220g and a variance of 100g. Find the probability that a pear that is picked at random has a mass

Hence, find the value of h such that 90% of the pears weigh more than h g.

Solution:

(a)

$\begin{array}{l}P\left(X>230\right)\\ =P\left(Z>\frac{230-220}{10}\right)\leftarrow \boxed{\begin{array}{l}\text{Convert to standard normal}\\ \text{distribution using}Z=\frac{X-\mu }{\sigma }\end{array}}\\ =P\left(Z>1\right)\\ =0.1587\end{array}$

(b)

$\begin{array}{l}P\left(210<X<225\right)\\ =P\left(\frac{210-220}{10}<Z<\frac{225-220}{10}\right)\leftarrow \boxed{\begin{array}{l}\text{Convert to standard normal}\\ \text{distribution using}Z=\frac{X-\mu }{\sigma }\end{array}}\\ =P\left(-1<Z<0.5\right)\\ =1-P\left(Z>1\right)-P\left(Z>0.5\right)\\ =1-0.1587-0.3085\\ =0.5328\end{array}$

$\begin{array}{l}\frac{h-220}{10}=-0.4602\\ h-220=-4.602\\ h=215.4\end{array}$