Expected Outcomes
When you roll a dice or draw from a bag, you can’t guarantee the exact result — but you can predict what’s likely to happen over many trials. That’s what expected outcomes are all about, and they’re used everywhere from game design to business forecasting.
The Core Concept
The Formula
The expected outcome is the predicted number of times an event will happen if you repeat an experiment many times.
It’s a prediction, not a guarantee — but the more trials you run, the closer your actual results tend to get to the expected outcome.
Key Terms
- Trial: One run of the experiment (e.g., one coin flip).
- Outcome: The result of a single trial.
- Event: The specific outcome you’re interested in.
- Fair: Every possible outcome is equally likely.
- Probability: A number between 0 (impossible) and 1 (certain) measuring how likely an event is.
$P(\text{Event}) = \dfrac{\text{Favourable outcomes}}{\text{Total outcomes}}$
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).
- Number of trials $= 200$.
- Expected heads $= \dfrac{1}{2} \times 200 = \mathbf{100}$.
Example 2: Dice Rolls
A fair six-sided dice is rolled 180 times. How many times would you expect to roll a number greater than 4?
- Favourable outcomes: {5, 6} → 2 outcomes.
$P(\text{> 4}) = \dfrac{2}{6} = \dfrac{1}{3}$. - Number of 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 chosen at random and replaced. This is done 150 times. How many times would you expect to choose a blue counter?
- Total counters $= 5 + 3 + 2 = 10$.
$P(\text{Blue}) = \dfrac{3}{10}$. - Number of trials $= 150$.
- Expected $= \dfrac{3}{10} \times 150 = \mathbf{45}$.
Tutor Insights
🤔 Common Misunderstandings
- Expected outcome = guaranteed outcome. It isn’t! If you flip a coin 10 times, you expect 5 heads — but you might get 4 or 7. The more trials you run, the closer actual results tend to get to the expected value.
- Non-whole-number results. Rolling a dice 50 times gives expected 6s of $\frac{1}{6} \times 50 \approx 8.33$. This just means you’d average about 8 or 9 sixes across many sets of 50 rolls.
📝 Common Exam Mistakes
- Forgetting to multiply by the number of trials — students often stop at the probability step.
- Miscounting outcomes when calculating the initial probability (always double-check your fraction).
- Misreading inequalities: “greater than 4” means {5, 6}, not {4, 5, 6}. “Less than or equal to 3” means {1, 2, 3}.
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 contains 4 red, 6 green, and 5 yellow sweets. A sweet is picked at random and replaced. This is done 300 times. How many times would you expect to pick a red sweet?
- Charlie rolls a fair six-sided dice 90 times. He wins a prize if he rolls a 6. How many prizes would you expect him to win?
- There are 50 students in a year group. The probability that a student chosen at random owns a pet is $\frac{3}{5}$. How many students would you expect to own a pet?
- A box contains 20 chocolates: 15 milk and 5 dark. Sarah takes a chocolate and replaces it each time. If she does this 40 times, how many times would you expect her to pick a dark chocolate?
Show Answers
- Odd numbers: {1,3,5,7} → 4 outcomes. $P(\text{Odd}) = \frac{4}{8} = \frac{1}{2}$. 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 prizes $= \frac{1}{6} \times 90 = \mathbf{15}$.
- Expected $= \frac{3}{5} \times 50 = \mathbf{30}$ students.
- $P(\text{Dark}) = \frac{5}{20} = \frac{1}{4}$. Expected $= \frac{1}{4} \times 40 = \mathbf{10}$.
FAQs
Q: Is the expected outcome always a whole number?
A: Not necessarily. $\frac{1}{6} \times 10 = 1.66\ldots$ — this just means you’d average about 1 or 2 sixes across many sets of 10 rolls. GCSE Foundation questions usually produce whole numbers, but it’s fine if they don’t.
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 repeat, the closer your actual results tend to get to the expected outcome. Over a small number of trials, results can vary a lot.
Q: What if the events aren’t equally likely?
A: The formula still works! Once you know the correct probability of the event (which may come from data or be given to you), you multiply it by the number of trials as usual. The calculation method is the same.
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