# Question Video: Determining Probabilities for Normal Distribution Mathematics

Let π be a normal random variable. Find the π(π > π + 0.71π).

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### Video Transcript

Let π be a normal random variable. Find the probability that π is greater than π plus 0.71π.

Remember the graph of a curve representing the normal distribution with a mean of π and a standard deviation of π is bell-shaped and symmetric about the mean. And the total area under the curve is 100 percent or one.

A sketch of the curve can be a really useful way to help decide how to answer a problem about normally distributed data. In this case, weβre looking to find the probability that π is greater than π plus 0.71π; thatβs this shaded region. We know it must sit above the mean in our bell curve because the standard deviation canβt be negative.

Once weβve established this, the next step with most normal distribution questions is to calculate the π-value. This is a way of scaling our data or standardising it in what becomes a standard normal distribution. Once we complete this step, we can work from a single standard normal table.

Now, it doesnβt really matter that we havenβt got a numerical value for the mean π or the standard deviation π of this data set. Letβs see what happens when we substitute everything we know into our formula for the π-value. Our value for π is π plus 0.71π and then we subtract π and we divide through by π. π minus π is zero.

So our formula simplifies somewhat to 0.71π all divided by π. However, we can simplify a little further by dividing through by π. And we get π is equal to 0.71. So weβre looking to find the probability that π is greater than 0.71 since in the original question, it was asking us to find the probability that π is greater than π plus 0.71π.

Our standard normal table though only gives probabilities between zero and π. In this case, thatβs this side of the curve. So we find the probability that π is greater than 0.71 by subtracting the probability that itβs less than 0.71 from one because we said that the area under the curve is 100 percent or one whole.

Looking up a π-value of 0.71 in our standard normal table and we can see that the probability that π is less than 0.711 is 0.7611. That means the probability that π is greater than 0.71 is one minus 0.7611; thatβs 0.2389.

That means the probability that π is greater than π plus 0.71π is equal to 0.2389.