Probability
Discrete Probability Distributions worksheets
Use valid finite probability tables to find missing entries, event probabilities, expected value, and population variance.
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Normal Distribution worksheets
Standardize normal values, find below, above, and between probabilities, and invert the model for percentiles and central intervals.
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Write the normal model first. Standardize probability cutoffs before using Phi, and use inverse normal for requested cutoffs. Keep full precision during calculation, then round probabilities to four decimals and cutoffs to two decimals.
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Each problem starts with a complete normal model, including a positive standard deviation. Learners standardize printed cutoffs exactly, shade the matching tail or interval, and choose cumulative, survival, difference, or inverse-normal work to match the question.
Learning objective
What this activity builds
Connect a stated normal model to z-scores, cumulative probability, upper tails, differences of cumulative probabilities, and inverse-normal cutoffs.
Worked example
For X normal with mean 50 and standard deviation 10, find the cutoff c with P(X < c) = 0.90.
Phi^-1(0.90) ≈ 1.2815516, so c = 50 + 10(1.2815516) ≈ 62.82.
The unknown is a cutoff, so invert the cumulative probability first and then undo standardization. Checking (62.82 − 50)/10 ≈ 1.282 gives a left-tail probability of about 0.9001; the small difference is rounding. About 10% lies above the cutoff.
Teaching notes
Make the practice count
- Sketch and shade the requested region before calculating.
- For an upper tail, use a survival calculation when available to avoid subtracting two nearly equal numbers.
- For a central interval, split the probability outside the interval equally between the two tails.
Normal distribution worksheets: probabilities and inverse cutoffs
Write the mean and positive standard deviation before using the normal model. For a value x, standardize with z = (x − mean)/standard deviation. Phi(z) is the area to the left; upper tails use 1 − Phi(z), and intervals use the upper cumulative probability minus the lower one. Normal probabilities describe an assumed model, not a guarantee about every observed data set.
For example, when the mean is 50 and the standard deviation is 10, P(40 < X < 60) = Phi(1) − Phi(−1) ≈ 0.8413 − 0.1587 = 0.6826 using rounded table entries (0.6827 from full precision). A percentile reverses this workflow: a 90th-percentile cutoff uses Phi^-1(0.90), not Phi(0.90). For a central 90% interval, the remaining 10% splits into 5% per tail, so use the 0.05 and 0.95 quantiles.
The worksheets support below, above and between probabilities, inverse percentiles and central intervals. Begin with shaded sketches, then mix area and cutoff tasks to assess method choice. Teachers can print Letter or A4 with a separate worked key. Keep full precision until the requested final rounding; broader model selection and nonnormal inference require other resources.
Is standard deviation the same as variance in the normal model?
No. Here the scale parameter is the positive standard deviation. If a course writes N(μ, σ²), its second parameter is variance; take its square root before using z = (x − μ)/σ.
Which area goes into inverse normal for a top-10% cutoff?
Use cumulative area 0.90 because 90% lies below the cutoff and 10% above it. For mean 50 and standard deviation 10 this gives approximately 62.82.
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Z-Scores worksheets
Calculate z-scores, recover original values, and compare relative positions across data sets with positive standard deviations.
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