Expected Frequency
Learn to calculate expected frequency for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-07). Expected frequency is how many times an event should happen when an experiment is repeated — the number of trials multiplied by the probability, \(np\).
You will find the probability from equally likely outcomes, a percentage or a relative frequency, multiply by the number of trials, and work backwards to find either one — useful for estimating how many people will win a prize or how many shots a player will make in a season.
Theory
Expected frequency is how many times an event is expected to happen in \(n\) trials: \(n\times p\). This Year 12 Standard 1 (NSW) guide covers dice, percentages and relative frequencies, and working backwards to find the number of trials or the probability.
The expected frequency of an event is how many times you expect it to happen when an experiment is repeated \(n\) times. Multiply the number of trials \(n\) by the probability \(p\) that the event occurs each time.
Find \(p\) first: from equally likely outcomes (a fair die: \(P(6)=\dfrac{1}{6}\)), from a given decimal or percentage (\(5\%=0.05\)), from “\(1\) in \(20\)” (\(p=\dfrac{1}{20}\)), or from a relative frequency.
An expected frequency is what happens on average in the long run. The actual count will usually be close to it, not exactly equal. If each success is worth points, find the expected number of successes, then multiply by the points. For example, \(50\) shots at \(p=0.6\), worth \(3\) points each: \(50\times0.6=30\) made, \(30\times3=90\) points.
Multiply the number of trials by the probability:
where \(n\) is the number of times the experiment is repeated and \(p\) is the probability the event occurs each time. Working backwards:
Shorthand: \(E(\text{event})\) means the expected number, e.g. \(E(6)=300\times\frac{1}{6}=50\).
How to find an expected frequency
- Find the probability \(p\) of the event (as a fraction or decimal).
- Identify the number of trials \(n\).
- Multiply \(n\times p\). If \(n\) or \(p\) is the unknown, divide instead.
Find each probability, then multiply by \(n=240\).
| \(\text{(i) }P(5)\) | \(=\) | \(\dfrac{1}{6}\) |
| \(\text{expected}\) | \(=\) | \(240\times\dfrac{1}{6}=40\) |
| \(\text{(ii) }P(>4)\) | \(=\) | \(\dfrac{2}{6}=\dfrac{1}{3}\ (5\text{ or }6)\) |
| \(\text{expected}\) | \(=\) | \(240\times\dfrac{1}{3}=80\) |
About \(40\) fives and \(80\) numbers greater than \(4\).
Write the percentage as a decimal, then use \(n\times p\).
| \(p\) | \(=\) | \(85\%=0.85\) |
| \(\text{expected}\) | \(=\) | \(n\times p\) |
| \(\) | \(=\) | \(60\times0.85\) |
| \(\) | \(=\) | \(51\) |
About \(51\) seeds are expected to germinate.
(ii) A spinner is expected to land on yellow \(45\) times in \(180\) spins. Find \(P(\text{yellow})\).
Use expected \(=n\times p\) and divide.
| \(\text{(i) }30\) | \(=\) | \(n\times0.15\) |
| \(n\) | \(=\) | \(\dfrac{30}{0.15}=200\) |
| \(\text{(ii) }45\) | \(=\) | \(180\times p\) |
| \(p\) | \(=\) | \(\dfrac{45}{180}=0.25\) |
(i) \(200\) games (ii) \(P(\text{yellow})=0.25\)
Use the relative frequency as the estimate of \(p\), then multiply by the new \(n\).
| \(p\) | \(\approx\) | \(\dfrac{18}{300}=0.06\) |
| \(\text{expected}\) | \(=\) | \(n\times p\) |
| \(\) | \(=\) | \(2000\times0.06\) |
| \(\) | \(=\) | \(120\) |
About \(120\) seedlings are expected not to survive.
Common pitfalls
Frequently asked questions
What is expected frequency?
Expected frequency is the number of times an event is expected to happen when an experiment is repeated. It is found by multiplying the number of trials by the probability of the event.
What is the formula for expected frequency?
Expected frequency = n times p, where n is the number of trials and p is the probability of the event each time. For 240 rolls of a die, the expected number of fives is 240 times 1 sixth = 40.
How do you find expected frequency with a percentage?
Change the percentage to a decimal, then multiply by the number of trials. If a packet says 85 percent of seeds germinate and 60 are planted, the expected number is 60 times 0.85 = 51.
How do you find the number of trials from the expected frequency?
Divide the expected number by the probability. To expect 30 wins when p = 0.15, play 30 divided by 0.15 = 200 games.
Will the actual result equal the expected frequency?
Usually not exactly. The expected frequency is an average over the long run, so the actual count will normally be close to it but can be a little more or less.
Can you use relative frequency to find expected frequency?
Yes. Use the relative frequency from past results as the probability, then multiply by the new number of trials. If 18 of 300 seedlings died, about 2000 times 0.06 = 120 of 2000 are expected to die.