Calculating Expected Outcomes
From predicting how many prize tickets will win at a school fair to testing whether a game is fair, expected outcomes let you use probability to forecast what will happen over many tries — a key skill for GCSE Maths and real life.
The Core Formula
The Formula
The expected outcome is the predicted average number of times an event will happen over many repetitions. It’s not a guarantee — it’s your best estimate.
Example: A fair coin, flipped 100 times. $P(\text{Head}) = \frac{1}{2}$. Expected heads $= \frac{1}{2} \times 100 = 50$.
Key Terms
- Trial: One run of the experiment (one flip, one roll).
- Outcome: The result of a single trial.
- Event: The specific outcome you’re interested in.
- Fair: All outcomes are equally likely.
- Probability: Always between 0 (impossible) and 1 (certain).
$P(\text{Event}) = \dfrac{\text{Favourable outcomes}}{\text{Total outcomes}}$
Worked Examples
Example 1: Coin Flips
A fair coin is flipped 200 times. How many heads would you expect?
- $P(\text{Head}) = \dfrac{1}{2}$ (1 head out of 2 outcomes).
- Trials $= 200$.
- Expected heads $= \dfrac{1}{2} \times 200 = \mathbf{100}$.
Example 2: Dice Rolls
A fair die is rolled 180 times. How many times would you expect to roll a number greater than 4?
- Favourable outcomes: {5, 6} → 2 outcomes.
$P(\text{>4}) = \dfrac{2}{6} = \dfrac{1}{3}$. - Trials $= 180$.
- Expected $= \dfrac{1}{3} \times 180 = \mathbf{60}$.
Example 3: Coloured Counters
A bag contains 5 red, 3 blue, and 2 yellow counters. A counter is drawn and replaced. This is done 150 times. How many times would you expect to pick blue?
- Total counters $= 5+3+2 = 10$.
$P(\text{Blue}) = \dfrac{3}{10}$. - Trials $= 150$.
- Expected $= \dfrac{3}{10} \times 150 = \mathbf{45}$.
Tutor Insights
🤔 Common Misunderstandings
- Expected outcome = guaranteed outcome. Flipping a coin 10 times doesn’t guarantee exactly 5 heads. The expected outcome is a long-run average — actual results vary, especially over few trials.
- Non-whole results. $\frac{1}{6} \times 50 = 8.33\ldots$ is fine — it means you’d average about 8 or 9 sixes over many sets of 50 rolls.
📝 Common Exam Mistakes
- Forgetting to multiply by the number of trials — stopping after calculating the probability.
- Miscounting favourable or total outcomes — always recheck your initial fraction.
- Misreading “greater than 4” as {4, 5, 6} instead of {5, 6}. Check your inequality carefully.
Practice Questions
- A fair spinner has 8 equal sections numbered 1–8. It is spun 240 times. How many times would you expect it to land on an odd number?
- A bag has 4 red, 6 green, and 5 yellow sweets. A sweet is picked and replaced 300 times. How many times would you expect to pick red?
- Charlie rolls a fair die 90 times, winning a prize on a 6. How many prizes would you expect him to win?
- The probability a randomly chosen student in a year group of 50 owns a pet is $\frac{3}{5}$. How many students would you expect to own a pet?
Show Answers
- Odd numbers: {1,3,5,7} → 4 out of 8. $P = \frac{1}{2}$. Expected $= \frac{1}{2} \times 240 = \mathbf{120}$.
- Total sweets $= 15$. $P(\text{Red}) = \frac{4}{15}$. Expected $= \frac{4}{15} \times 300 = \mathbf{80}$.
- $P(\text{Six}) = \frac{1}{6}$. Expected $= \frac{1}{6} \times 90 = \mathbf{15}$ prizes.
- Expected $= \frac{3}{5} \times 50 = \mathbf{30}$ students.
FAQs
Q: Is the expected outcome always a whole number?
A: Not necessarily. $\frac{1}{6} \times 10 = 1.66\ldots$ just means you’d average about 1 or 2 sixes over many sets of 10 rolls. GCSE Foundation questions usually give whole-number results, but decimals are perfectly valid answers.
Q: Does “expected outcome” mean it will definitely happen exactly that many times?
A: No — it’s a prediction, not a guarantee. The more trials you run, the closer your actual results tend to be to the expected outcome. Over just a few trials, results can vary a lot.
Q: What if the events aren’t equally likely?
A: The formula still works. Once you have the correct probability of the event (which may be given to you or require data), multiply it by the number of trials as usual.
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