Choosing the wrong tail is one of the most common errors in normal distribution questions - especially when the question says 'within', 'outside' or 'more than'.
In short: One-tailed and two-tailed probabilities describe whether the region of interest lies at one end of a distribution or at both ends. A one-tailed question asks about more than or less than a value, while a two-tailed one asks about outside or within symmetric limits. Sketch the curve, shade the region, then choose the bounds.
When you'd use this
The question says "more than" or "less than" (one-tailed) rather than "within" or "outside" (two-tailed).
Working out P(|X − μ| > k), which needs doubling one tail, not just reading off the calculator.
Setting up the correct region for a hypothesis test before finding its p-value.
Avoiding the single most common error in normal distribution exam questions.
Steps (the same on every model)
Sketch the normal curve and shade the region the question is asking about before touching the calculator.
Left tail P(X < a): normalcdf(−1E99, a, μ, σ).
Right tail P(X > a): normalcdf(a, 1E99, μ, σ).
Between P(a < X < b): normalcdf(a, b, μ, σ).
Outside P(X < a or X > b): 1 − normalcdf(a, b, μ, σ) - or compute each tail separately and add.
Symmetric two-tail: if P(|X − μ| < k) = 0.9, find the upper bound using invNorm(0.95, μ, σ) - the central 90% leaves 5% in each tail.
For inverse: always convert to the area to the LEFT before using invNorm.
Tip: Draw the curve first, every time. Most errors come from shading the wrong region mentally without a sketch.
Here's a real IB-style question that uses exactly this technique.
MediumCalculatorPaper 2[3 marks]
X ~ N(100, 15²). Find P(|X − 100| > 20), to 3 significant figures.
0.182
Mark it
Correct3 / 3 marks
Worked solution & mark scheme:
M1 Find P(X > 120) and double it (by symmetry)
A1 0.182
Common questions
When would I need to decide between one-tailed and two-tailed probabilities in IB Maths?
Choosing the wrong tail is one of the most common errors in normal distribution questions - especially when the question says 'within', 'outside' or 'more than'. The question says "more than" or "less than" (one-tailed) rather than "within" or "outside" (two-tailed). Working out P(|X − μ| > k), which needs doubling one tail, not just reading off the calculator.
How do I decide between one-tailed and two-tailed probabilities?
1. Sketch the normal curve and shade the region the question is asking about before touching the calculator. 2. Left tail P(X < a): normalcdf(−1E99, a, μ, σ). 3. Right tail P(X > a): normalcdf(a, 1E99, μ, σ). 4. Between P(a < X < b): normalcdf(a, b, μ, σ). 5. Outside P(X < a or X > b): 1 − normalcdf(a, b, μ, σ) - or compute each tail separately and add. 6. Symmetric two-tail: if P(|X − μ| < k) = 0.9, find the upper bound using invNorm(0.95, μ, σ) - the central 90% leaves 5% in each tail. 7. For inverse: always convert to the area to the LEFT before using invNorm.
What should I watch out for when I decide between one-tailed and two-tailed probabilities?
Draw the curve first, every time. Most errors come from shading the wrong region mentally without a sketch.
How are marks awarded when I decide between one-tailed and two-tailed probabilities in an IB exam?
In the worked example on this page (3 marks, Paper 2), the marks are: M1: Find P(X > 120) and double it (by symmetry); A1: 0.182.
