Many measurements pile up symmetrically around a mean, forming a bell-shaped curve.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The 68-95-99.7 rule
About 68% of values lie within one standard deviation of the mean, 95% within two, 99.7% within three.
Conditional probability
P(A given B) asks the probability of A among only the cases where B happened.
It restricts the pool
The denominator changes from everything to just the B cases. That is the whole idea.
The normal distribution
Many measurements cluster symmetrically around a mean in a bell shape. About 68% fall within one standard deviation, 95% within two, and 99.7% within three. Those three numbers do most of the work in practice.
A z-score measures unusualness
z = (value − mean)/standard deviation says how many standard deviations from the mean a value sits. It puts any measurement on a common scale, so a height and a test score become directly comparable.
Conditional probability
P(A | B) is the probability of A given that B has happened. Restricting attention to the cases where B occurred changes the denominator, which is why conditional probabilities can differ wildly from unconditional ones.
Independence has a precise test
A and B are independent when P(A | B) = P(A) — knowing B tells you nothing about A. Assuming independence when it does not hold is the most consequential error in applied probability, and it caused real financial disasters.
Step 2: Try It Yourself
Tap and try it out.
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Measured value13 things were measured in all.
The tallest column is at 5, with 5 marks.
Step 3: Watch an Example
One step at a time.
Watch Ravi Use the Rule
Heights have mean 170 cm and standard deviation 8 cm.
- Step 1
One standard deviation each way spans 162 to 178 cm.
Step 4: Your Turn
Practice makes it stick.
The Range
Problem 1 of 2
Mean 50, standard deviation 5. About what percent lie between 45 and 55?
percent
The Wider Range
Problem 2 of 2
Mean 50, standard deviation 5. About what percent lie between 40 and 60?
percent
Spread and Chance
1 of 4
What percent lies within one standard deviation?
2 of 4
What percent lies within two standard deviations?
3 of 4
20 students, 8 play sport. 5 of those 8 also play music. Out of 8, how many play music?
4 of 4
Does conditioning on B change the denominator?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Mean 100, standard deviation 15. About what percent lie between 85 and 115?
Question 2 of 2
What does conditional probability change?
What You Learned
- Many measurements form a bell curve around the mean.
- About 68%, 95%, and 99.7% lie within one, two, and three deviations.
- Conditional probability restricts the pool of cases.