True or False? Probabilities can only be expressed as fractions .

False. Probabilities can be expressed as fractions, decimals or percentages .

What does it mean when two events are mutually exclusive .

Mutually exclusive events are events that cannot both happen at the same time. E.g. when a dice is rolled, you cannot obtain a 6 and a 3 at the same time, therefore they are mutually exclusive events.

What is a sample space in the context of probability?

A sample space is the set of all possible outcomes of an experiment. It may be a list or a grid showing all the possible outcomes.

True or False? You can always use a sample space grid to represent all possible outcomes .

False. When combining more than two things , you may need to go back to listing outcomes instead of using a grid.

How do you calculate probabilities using a sample space ?

Count the possibilities you want , then divide by the total number of possibilities in the sample space. (Note: this only works if every possibility in the sample space is equally likely .)

What are systematic listing strategies when calculating probabilities?

Systematic listing strategies are ways of writing out all possible combinations or arrangements of items in an organised way, without missing any.

How could you use a systematic listing strategy to list all the possible ways to arrange the letters A, B and C?

For listing arrangements of letters A, B, C: Fix A first, and list ABC, ACB Fix B next, and list BAC, BCA Fix C last, and list CAB, CBA The six arrangements are ABC, ACB, BAC, BCA, CAB and CBA.

Why is it better to use a systematic listing strategy, rather than an unsystematic list?

Systematic listing strategies help ensure completeness and avoid missing any possibilities , whereas unsystematic listing risks missing options.

Define relative frequency in the context of probability.

In probability, relative frequency (also called experimental probability) is used to estimate probabilities from experimental data .

True or False? In order to estimate relative frequency in an experiment, each trial must have an equal chance of 'success' .

True. In order to estimate relative frequency in an experiment, each trial must have an equal chance of 'success' . I.e. the trials are independent. E.g. when flipping a coin, getting a 'head' on one flip will not affect the chances of getting a 'head' on another flip. The trials are independent of each other and have an equal chance of success, therefore they can be used to estimate the relative frequency of getting a 'head'.

True or False? Calculating a relative frequency lets you know the exact probability .

False. Relative frequency only provides an estimate for the true (or theoretical) probability .

Define expected frequency in the context of probability.

Expected frequency refers to the number of times a particular outcome is expected to occur when repeating a trial numerous times.

True or False? If an experiment is repeated many times, the number of occurrences of an event will always be exactly equal to the expected frequency .

False. The number of occurrences tends to average out to the expected frequency over many repetitions, but may not be exactly equal .

True or false? Relative frequency and expected frequency can both take on any positive value .

False. Expected frequency can in theory take on any positive value . But relative frequency is an estimate of a probability , so must be between 0 and 1 .

True or false? A larger number of trials should give a more accurate result for relative frequency than a smaller number of trials.

True. Usually a relative frequency calculated from a large number of trials is more reliable than one calculated from a small number of trials.

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What does the notation $P (A)$ mean, in the context of probability?

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What does the notation $P (A)$ mean, in the context of probability?
$P (A)$ is the notation for the probability of event A occurring.
True or False?
Probabilities can only be expressed as fractions.
False.
Probabilities can be expressed as fractions, decimals or percentages.
What does it mean when two events are mutually exclusive.
Mutually exclusive events are events that cannot both happen at the same time.
E.g. when a dice is rolled, you cannot obtain a 6 and a 3 at the same time, therefore they are mutually exclusive events.
What is the complement of event A?
The complement of event A is the event where event A does not happen.
It is denoted by $A^{'}$ .
State an equation connecting $P (A)$ and $P (A^{'})$ .
The key equation connecting $P (A)$ and $P (A^{'})$ is $P (A) + P (A^{'}) = 1$ .
It can be rearranged as $P (A^{'}) = 1 - P (A)$ or $P (A) = 1 - P (A^{'})$ .
True or False?
There are only two possibilities in a lottery, 'win' and 'lose'.
Therefore, the probability of winning is $\frac{1}{2}$ .
False.
The outcomes 'win' and 'lose' are not equally likely, so the probability of winning is not $\frac{1}{2}$ .
What is a sample space in the context of probability?
A sample space is the set of all possible outcomes of an experiment.
It may be a list or a grid showing all the possible outcomes.
True or False?
You can always use a sample space grid to represent all possible outcomes.
False.
When combining more than two things, you may need to go back to listing outcomes instead of using a grid.
How do you calculate probabilities using a sample space?
Count the possibilities you want, then divide by the total number of possibilities in the sample space.
(Note: this only works if every possibility in the sample space is equally likely.)
What are systematic listing strategies when calculating probabilities?
Systematic listing strategies are ways of writing out all possible combinations or arrangements of items in an organised way, without missing any.
How could you use a systematic listing strategy to list all the possible ways to arrange the letters A, B and C?
For listing arrangements of letters A, B, C:
- Fix A first, and list ABC, ACB
- Fix B next, and list BAC, BCA
- Fix C last, and list CAB, CBA
The six arrangements are ABC, ACB, BAC, BCA, CAB and CBA.
Why is it better to use a systematic listing strategy, rather than an unsystematic list?
Systematic listing strategies help ensure completeness and avoid missing any possibilities, whereas unsystematic listing risks missing options.
Define relative frequency in the context of probability.
In probability, relative frequency (also called experimental probability) is used to estimate probabilities from experimental data.
What is the equation used to calculate relative frequency?
Relative frequency can be calculated using the following equation:
$Relative frequency = \frac{Number of successful outcomes}{Total number of trials}$
True or False?
In order to estimate relative frequency in an experiment, each trial must have an equal chance of 'success'.
True.
In order to estimate relative frequency in an experiment, each trial must have an equal chance of 'success'. I.e. the trials are independent.
E.g. when flipping a coin, getting a 'head' on one flip will not affect the chances of getting a 'head' on another flip.
The trials are independent of each other and have an equal chance of success, therefore they can be used to estimate the relative frequency of getting a 'head'.
True or False?
Calculating a relative frequency lets you know the exact probability.
False.
Relative frequency only provides an estimate for the true (or theoretical) probability.
Define expected frequency in the context of probability.
Expected frequency refers to the number of times a particular outcome is expected to occur when repeating a trial numerous times.
How do you calculate the expected frequency of an event with probability $p$ , if the experiment is repeated $n$ times.
To calculate the expected frequency of an event with probability $p$ , if the experiment is repeated $n$ times is, you must multiply the number of trials by the probability that the event will occur.
$Expected frequency = n p$ .
True or False?
If an experiment is repeated many times, the number of occurrences of an event will always be exactly equal to the expected frequency.
False.
The number of occurrences tends to average out to the expected frequency over many repetitions, but may not be exactly equal.
True or false?
Relative frequency and expected frequency can both take on any positive value.
False.
Expected frequency can in theory take on any positive value.
But relative frequency is an estimate of a probability, so must be between 0 and 1.
True or false?
A larger number of trials should give a more accurate result for relative frequency than a smaller number of trials.
True.
Usually a relative frequency calculated from a large number of trials is more reliable than one calculated from a small number of trials.

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