Relative Frequency and Probability

Relative frequency and probability is about measuring chance, either by counting equally likely outcomes or by running an experiment.

• The probability scale from 0 (impossible) to 1 (certain), written as fractions, decimals or percentages.
• The sample space: the set of all possible outcomes of an experiment, and how many outcomes there are.
• Theoretical probability for equally likely outcomes.
• Complementary events: $P(\text{not }A)=1-P(A)$
• Two-stage experiments shown with tables, lists or tree diagrams, with and without replacement.
• Relative frequency as an estimate of probability, and the law of large numbers.
• Expected frequency $=np$.
• How statistics and probabilities are used in decisions by governments, businesses and the media.

• An experiment (or trial) is an action with uncertain results, such as rolling a die.
• An outcome is one possible result.
• The sample space is the set of all outcomes: for a die, $\{1,2,3,4,5,6\}$.
• An event is a set of outcomes we're interested in, such as "rolling an even number".

Every probability is between 0 and 1:

A probability of $\frac{1}{4}$ can also be written as 0.25 or 25%.

When all outcomes are equally likely:

Example. A letter is chosen at random from the word PROBABILITY (11 letters). There are two Bs and two Is, so $P(\text{B})=\frac{2}{11}$ and $P(\text{vowel})=\frac{4}{11}$ (O, A, I, I).

Complementary events

The complement of an event is everything else in the sample space: "not A". Because one of them must happen:

Example. If the probability a flight is on time is 0.83, the probability it is not on time is $1-0.83=0.17$.

Tables and lists

When two things happen together (two coins, a coin and a die, two spinners), list every combined outcome in a table.

Worked example. A coin is tossed and a die is rolled.

There are $2\times 6=12$ equally likely outcomes. Head and an even number: H2, H4, H6, so

The number of outcomes for a two-stage experiment is the number of outcomes in the first stage times the number in the second stage.

Tree diagrams

A tree diagram shows the stages one after the other. Write the probability on each branch.

• Multiply along a path to find the probability of that combined outcome.
• Add the probabilities of different paths that fit the event.
• The probabilities at the ends of all paths add to 1, which is a useful check.

With and without replacement

Worked example. A bag contains 2 red and 3 white balls. Two balls are drawn at random.

With replacement (the first ball is put back), the probabilities are the same on both draws:

Without replacement, there is one ball fewer on the second draw, and one fewer of the colour already drawn:

For "one of each colour" without replacement, add both orders:

Check: $P(RR)=\frac{2}{5}\times \frac{1}{4}=\frac{1}{10}$ and $\frac{1}{10}+\frac{3}{10}+\frac{3}{5}=1$

When outcomes aren't equally likely, or we don't know the probabilities, we estimate them from data:

Example. A quality check finds 14 faulty light globes in a sample of 500. The relative frequency of a faulty globe is $\frac{14}{500}=0.028$, so the estimated probability that a globe is faulty is 0.028 (2.8%).

The law of large numbers

The more trials there are, the closer the relative frequency usually gets to the true probability. Ten coin tosses might give 7 heads (0.7), but 10 000 tosses will almost always give very close to 0.5. That is why estimates from large samples are more reliable than those from small ones.

The expected frequency of an event over $n$ trials is

where $p$ is the probability of the event on each trial.

Example. Using the relative frequency of 0.028 above, a factory making 20 000 globes would expect about $20000\times 0.028=560$ faulty globes.

Expected frequency is an average: the actual number will vary a little from batch to batch.

Is a game fair?

Compare the expected winnings with the cost of playing. If players expect to get back less than they pay, the game favours the organiser. Find the probability of winning, the expected number of wins, then compare money in and money out.

Probabilities and statistics drive many real decisions:

• Insurance: premiums are set from the estimated probability of a claim. Young drivers usually pay more for car insurance because, as a group, they have more accidents.
• Weather forecasts: a 70% chance of rain means that, on days with similar conditions, it rained on about 70% of them.
• Health and government: vaccination programs and road safety campaigns are planned using rates of illness and accidents.
• Media: headlines can mislead, for example by reporting that a risk "doubles" when it goes from 1 in a million to 2 in a million.

• Treating outcomes as equally likely when they aren't, such as the totals when two dice are rolled (7 is much more likely than 2).
• Forgetting to reduce the total on the second draw without replacement.
• Adding probabilities along a branch instead of multiplying.
• Including only one order (RW but not WR) for "one of each".
• Giving a relative frequency as the exact probability.
• Writing a probability greater than 1 or less than 0: check every answer.

• Show the sample space (a list, table or tree) when the question involves two stages; it often earns a mark on its own.
• Leave answers as simplified fractions unless asked for a decimal or percentage.
• For expected frequency, write $n\times p$ with the numbers substituted.
• For "is this fair?" or "should they…?" questions, finish with a clear conclusion supported by your numbers.

• Data analysis (Year 11): frequency tables give the counts used for relative frequency.
• Bivariate data analysis (Year 12): both topics use data to make predictions and judge how reliable they are.
• Managing money (Year 11): insurance and the cost of risk.