Repeat n times, expect n × P successes: If an event has probability P and a trial is repeated n times, the expected number of occurrences is n × P. It's the long-run average — it need not be a whole number.
IB-style question — basic expected number
The probability that it rains on any day in April is 0.3. Find the expected number of rainy days in the 30 days of April.
Step by step
- Expected number = n × P.
Final answer
9 rainy days expected.
Decimals are fine: An expected number like 7.5 is acceptable — it's an average over many repeats, not a count from one run.
Work out P, then multiply by n: Often the probability isn't given directly — find P first (from a sample space, proportion or table), then multiply by the number of trials n.
IB-style question — P then × n
A spinner has 8 equal sectors, 3 of which are red. The spinner is spun 40 times. Find the expected number of reds.
Step by step
- Probability of red on one spin.
- Expected number = n × P.
Final answer
15 reds expected.
Two clean steps: Always show the probability and the multiplication — both usually earn marks.
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Expected total = (per-trial average) × n: The same idea extends to an expected amount: if each trial has an average outcome (an expected value), then over n trials the expected total is n × (average per trial). It predicts the long-run total, not any single result.
IB-style question — expected winnings
In a game you win $5 with probability 0.2 and otherwise lose $1. Find the expected gain per game, then the expected total over 50 games.
Step by step
- Expected gain per game.
- Over 50 games.
Final answer
Expected gain $0.20 per game; about $10 over 50 games.
It's an average, not a promise: Expecting $10 doesn't mean you'll win exactly $10 — it's what you'd average over many repeats.