Discrete and binomial distributions
Use probability weights and recognise the assumptions behind a binomial model.
What you will learn
- Calculate an expected value.
- Use a binomial model for repeated trials.
Expected value is a weighted mean
For a discrete random variable, list each possible value and its probability. Probabilities must be non-negative and sum to one. The expected value is the sum of value multiplied by probability; it need not be a possible outcome.
An expectation describes a long-run average under repeated use of the model. It does not predict the exact result of the next trial.
Check the binomial assumptions
A binomial variable counts successes in a fixed number n of independent trials, each with the same success probability p and two classified outcomes. When sampling without replacement from a small population, constant p and independence usually fail.
Exactly zero successes means every trial fails, giving (1−p)ⁿ. At least one success is the complement, 1−(1−p)ⁿ.
Worked example
X takes values 0,1,3 with probabilities 0.2,0.5,0.3. Find E(X).
Show the worked solution
- Check the probabilities: 0.2+0.5+0.3=1.
- Multiply values by probabilities: 0,0.5,0.9.
- Add the weighted values.
Answer 1.4
Common mistakes
- Do not average the possible values without probability weights.
- The expected value does not have to be an integer.
What can you explain now?
For four independent trials with success probability 0.25, find the probability of no successes.
Compare with the explanation
0.31640625
All four trials must fail, so the probability is 0.75⁴.
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