5.5. Applications from Physics, Engineering, and Statistics http://www.ck12.org
Normal Probabilities
If you were told by the postal service that you will receive the package that you have been waiting for sometime
tomorrow, what is the probability that you will receive it sometime between 3:00 PM and 5:00 PM if you know that
the postal service’s hours of operations are between 7:00 AM to 6:00 PM?
If the hours of operations are between 7 AM to 6 PM, this means they operate for a total of 11 hours. The interval
between 3 PM and 5 PM is 2 hours, and thus the probability that your package will arrive is
P=11 hours2 hours = 0. 182
= 18 .2%So there is a probability of 18.2% that the postal service will deliver your package sometime between the hours of 3
PM and 5 PM (or during any 2−hour interval). That is easy enough. However, mathematically, the situation is not
that simple. The 11−hour interval and the 2−hour interval contain an infinite number of times. So how can one
infinity over another infinity produce a probability of 18.2%? To resolve this issue, we represent the total probability
of the 11−hour interval as a rectangle of area 1 (Figure 25).
Figure 25
Looking at the 2−hour interval, we can see that it is equal to 112 of the total rectangular area 1.This is why it is
convenient to represent probabilities as areas. But since areas can be defined by definite integrals, we can also define
the probability associated with an interval[a,b]by the definite integral
P=
∫b
a f(x)dx,wheref(x)is called theprobability density function(pdf). One of the most useful probability density functions is
thenormal curveor theGaussian curve(and sometimes thebell curve) (Figure 26). This function enables us to
describe an entire population based on statistical measurements taken from a small sample of the population. The
only measurements needed are the mean(μ)and the standard deviation(σ). Once those two numbers are known,
we can easily find the normal curve by using the following formula.