Expected Value and Variance, Gently
Understand averages of uncertain outcomes and how much they spread, with dice, games and decisions
A taste of a lesson
Option A pays 100 for sure. Option B pays 250 with 40 percent chance, else 0. Which is better?
Start with expected values. A is 100. B is 0.4 times 250, which is also 100. So on average they are equal, but B is far riskier: you get nothing 60 percent of the time. If this is a one off choice and losing would hurt, A is reasonable. If you face many similar, affordable choices, the risk averages out and either works. Try computing B's standard deviation: the variance is 0.4 times (250 minus 100) squared plus 0.6 times (0 minus 100) squared. What do you get?
Written by the teacher as an example. In your lesson the tutor answers your own questions, and like any AI it can be wrong.
What you will be able to do
- Compute expected values from tables of outcomes and probabilities
- Explain why the expected value is not the most likely outcome
- Use linearity of expectation to combine quantities
- Compute variance and standard deviation and combine them for independent quantities
- Weigh average outcomes against risk in simple decisions
Lesson plan
- 1 The long run average Compute expected values for simple games and decisions. Start
- 2 Combining expectations Use linearity to simplify calculations. Start
- 3 Measuring spread Compute variance and standard deviation. Start
- 4 Adding uncertain quantities Combine variances for independent quantities. Start
- 5 Why averages stabilise See how averaging reduces variance. Start
- 6 Risk and decisions Weigh expected value against variability. Start
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About this tutor
A beginner tutor on two ideas that underpin statistics, machine learning and everyday decisions under uncertainty. You will compute expected values for dice, games, insurance and simple business choices, see why the expected value is not the most likely outcome, and learn the rules that make expectations easy to combine. Then you will measure spread with variance and standard deviation, discover why averaging many independent results reduces variance, and think about when risk matters as much as the average. Each lesson uses small numbers and a short decision exercise.
Reviews
4.3
3 ratingsSample
- Jonas P.Sample
Clear and gentle. The 'variances add, standard deviations do not' rule saved me from a mistake at work.
- Mei-Ling C.Sample
I liked the honest discussion of risk versus average. Good foundation before machine learning.
- Aaliyah W.Sample
The game examples made expected value click immediately. I finally see why a die averages 3.5.
About the teacher
Probability for machine learning, plus forecasting and anomaly detection
9 tutors 361 lessons taught Sample
I teach probability the way machine learning uses it: random variables, likelihood, entropy and simulation. I also teach two applied areas where probability matters every day: time series forecasting and anomaly detection. My work background is in monitoring and forecasting for operational systems, where wrong alarms and missed incidents both have a cost. I teach through small simulations, coin and...
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