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Calculate probabilities for normal distributions

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Which standard-normal expression matches the event below?

\[X\sim N(40,9),\quad X>43\]
Year 12, VCE Units 3 & 4, Chapter 13, 4 difficulty levels, Calculator allowedBuild a worksheet

Normal probability calculations

A general normal probability is converted to a standard normal probability by standardising every boundary. Then use the matching tail, interval difference or two-tail sum. Use the 68–95–99.7% empirical rule to estimate normal probabilities

\[z=\frac{x-\mu}{\sigma}\]
  1. Standardise every boundary Subtract the mean and divide by the standard deviation \(\displaystyle x=43,\ \mu=40,\ \sigma=3\;\Rightarrow\;z=\frac{43-40}{3}=1\)
  2. Match the tail direction An event above a boundary uses the upper tail \(\displaystyle X>43\;\Rightarrow\;\Pr(Z>1)\)
  3. Subtract for an interval The probability between two boundaries is the upper less-than area minus the lower less-than area \(\displaystyle \Pr(-1<Z<1)\approx 0.84-0.16=0.68\)
  4. Add the two outside tails A two-tail probability is found by adding the lower and upper tails or complementing the central area \(\displaystyle \Pr(|Z|>2)\approx 1-0.95=0.05\)

Worked example

Use the 68–95–99.7% empirical rule to estimate the probability shown for the given normal distribution

\[\Pr(25<X<35),\quad X\sim N(30,25)\]
  1. \(\displaystyle z_1=\frac{25-30}{5}=-1\)Standardise the lower boundary using the mean 30 and standard deviation 5
  2. \(\displaystyle z_2=\frac{35-30}{5}=1\)Standardise the upper boundary in the same way
  3. \(\displaystyle \Pr(25<X<35)=\Pr(-1<Z<1)\approx 0.84-0.16\)For an interval, subtract the lower less-than area from the upper less-than area
  4. \(\displaystyle \Pr(25<X<35)\approx 0.68=68\%\)The empirical rule gives 0.68, or 68 percent, between one standard deviation below and above the mean

Common mistakes

Reversing the tail
✕\(\displaystyle X>43\;\Rightarrow\;\Pr(Z<1)\approx 0.84\)✓\(\displaystyle X>43\;\Rightarrow\;\Pr(Z>1)\approx 1-0.84=0.16\)
The distribution has mean 40 and standard deviation 3. An above-boundary event uses the upper tail, so the lower-tail area must be complemented. Use the 68–95–99.7% empirical rule to estimate normal probabilities
Using a raw boundary as z
✕\(\displaystyle z=43\)✓\(\displaystyle z=\frac{43-40}{3}=1\)
The distribution has mean 40 and standard deviation 3. A general normal boundary must be centred by subtracting the mean and scaled by dividing by the standard deviation
Dividing by the variance
✕\(\displaystyle z=\frac{43-40}{9}=\frac{1}{3}\)✓\(\displaystyle z=\frac{43-40}{3}=1\)
The distribution has mean 40 and standard deviation 3. Standardisation divides by the standard deviation, not the variance

Difficulty levels

introFind a one-tail probability for the standard normal distribution
standardStandardise two general-normal boundaries and calculate an interval probability
challengeCalculate a two-tail probability or use a non-integer standardisation value
examChoose the correct standardisation expression before finding a boundary standard score

Questions about this topic

Why is each boundary standardised?+

Standard normal probabilities are stated in standard-deviation units. To find a value's standard score, subtract the mean from the value and divide by the standard deviation

How is the probability between two boundaries found?+

Find the less-than area for each standard score, then subtract the lower area from the upper area

Why is an upper-tail probability a complement?+

The lower-tail and upper-tail areas together make 1. Therefore the area above a boundary is 1 minus the area below it

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Mathematical Methods, chapter 13: Normal distribution

In this chapter

Read normal parameters and describe their graph effectCalculate probabilities for normal distributionsCalculate a standard score or actual valueUse z-score boundaries to estimate normal-model percentages and countsCompare observations from different distributions using z-scoresInverse normal calculationsInfer a normal-model mean or standard deviationConditional probabilities with normal intervalsMixed probability applications with the normal distribution