Relative frequency and expected outcomes

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Lesson notes

Theoretical and experimental probability

Theoretical probability = number of favourable outcomes ÷ total equally likely outcomes. A fair die gives P(6) = \frac{1}{6}.
Relative frequency (experimental probability) = number of times the event happened ÷ total number of trials.
Probabilities lie between 0 (impossible) and 1 (certain), and all outcomes sum to 1: P(A) + P(\text{not } A) = 1.

Relative frequency and the law of large numbers

Over many independent trials with a stable underlying probability, relative frequency tends to settle near that probability. More trials generally reduce random variation, but do not guarantee a closer estimate at every step or remove systematic bias.
A spinner landing on red 18 times in 50 spins has relative frequency \frac{18}{50} = 0.36.

Expected outcomes

Expected number = probability × number of trials. If P(\text{red}) = 0.2 and there are 150 trials, expect 0.2 \times 150 = 30.
Expected values are averages; actual results vary.

Expected frequencies

Bias and fairness

If the relative frequencies are very different from equal values over many trials, the die or coin may be biased.
Compare experimental results to the theoretical probability to judge fairness.

Estimating probabilities

Use data (e.g. past weather) to estimate probability when outcomes are not equally likely.

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Practice questions

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1.What is the probability of an impossible event?
Easy
- A0
- B0.5
- C1
- D-1
2.What is P(\text{even}) when a fair die is rolled?
Easy
- A\frac{1}{6}
- B\frac{1}{3}
- C\frac{1}{2}
- D\frac{2}{3}
3.A coin lands on heads 30 times in 100 tosses. What is the relative frequency of heads?
Easy
- A0.3
- B0.7
- C30
- D0.5
4.If P(\text{rain}) = 0.35, what is P(\text{no rain})?
Easy
- A0.65
- B0.35
- C0.45
- D1.35
5.A spinner has P(\text{red}) = 0.2. How many reds do you expect in 150 spins?
Easy
- A15
- B20
- C30
- D75
6.A bag has 3 red and 7 blue counters. What is P(\text{red})?
Easy
- A\frac{3}{7}
- B\frac{3}{10}
- C\frac{7}{10}
- D\frac{1}{3}
7.In 200 trials an event occurs 50 times. What is the relative frequency?
Easy
- A0.25
- B0.5
- C4
- D0.05
8.For independent trials under unchanged conditions, why does a large sample generally give a more reliable probability estimate than a small sample?
Easy
- Arandom fluctuations in the relative frequency are typically smaller
- Bit is quicker
- Cit is always exact
- Dthere is no difference

More questions

9.Which cannot be a probability?
Easy
- A0
- B0.5
- C1.2
- D1
10.A biased die is rolled 200 times and a six comes up 60 times. What is the relative frequency of a six?
Medium
- A0.3
- B0.17
- C0.6
- D3
11.A fair coin is tossed 500 times. About how many tails do you expect?
Medium
- A100
- B250
- C500
- D50
12.The probability of a defective item is 0.02. In a batch of 2500, how many do you expect to be defective?
Medium
- A25
- B50
- C125
- D500
13.A spinner lands on blue 24 times in 80 spins. Estimate the probability of blue.
Medium
- A0.3
- B0.24
- C0.8
- D0.33
14.A die is rolled 600 times. The number of fours is 150. Is the die likely to be fair?
Medium
- Ano, the relative frequency 0.25 is far from \frac{1}{6}
- Byes
- Ccannot tell at all
- Dyes because 150 is a nice number
15.Which is the best estimate for the probability of a drawing pin landing point up if it did so 62 times in 100 throws?
Medium
- A0.62
- B0.38
- C0.5
- D6.2
16.If the probability of rain is 0.3 on each of 30 days, how many rainy days are expected?
Medium
- A9
- B3
- C10
- D21
17.A bag has 4 red and 6 blue counters. One blue counter is removed. What is the probability that a counter picked at random is blue?
Medium
- A\frac{5}{9}
- B\frac{1}{2}
- C\frac{5}{10}
- D\frac{4}{9}
18.A bag has 5 red, 3 blue and some green counters. P(\text{green}) = \frac{1}{3}. How many greens?
Medium
- A4
- B3
- C8
- D2
19.Two events A and B are mutually exclusive with P(A) = 0.3 and P(B) = 0.5. What is P(A \text{ or } B)?
Medium
- A0.8
- B0.15
- C0.2
- D1.5
20.The sum of the probabilities of all possible outcomes is:
Easy
- A0
- B1
- C100
- Ddepends
21.Expected number =
Easy
- Aprobability × trials
- Bprobability ÷ trials
- Ctrials − probability
- Dtrials ÷ probability
22.A probability of 1 means the event is certain.
Easy
True or false?
23.Relative frequency is based on experimental results.
Easy
True or false?
24.A fair coin lands heads in 3 of 5 independent tosses. This proves that its true probability of heads is 3/5.
Easy
True or false?
25.More trials generally give a better estimate of probability.
Easy
True or false?
26.Expected numbers always happen exactly.
Medium
True or false?
27.Which can be probabilities?
Medium
- A0
- B0.75
- C1
- D\frac{3}{8}
- E1.5
28.Which of these would suggest a die is biased?
Medium
- Amany more sixes than expected over thousands of rolls
- Ba few more sixes in ten rolls
- Ceach face about \frac{1}{6} over 6000 rolls
- Done face appearing 40% of the time over 1000 rolls
- Ethe die is red
29.A fair coin is tossed. What is P(\text{tails})?
Easy
- A\frac{1}{4}
- B\frac{1}{2}
- C1
- D2
30.A bag has 2 red, 3 green and 5 blue counters. What is P(\text{green})?
Easy
- A\frac{3}{10}
- B\frac{3}{7}
- C\frac{1}{3}
- D\frac{1}{10}
31.In 80 trials an event happens 20 times. What is the relative frequency?
Easy
- A0.25
- B0.2
- C4
- D0.75
32.A spinner is spun 200 times and lands on blue 54 times. Estimate the probability of blue, and the number of blues expected in 500 spins.
Medium

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