Define the term experiment in the context of probability.

In probability, an experiment is a repeatable activity that has a result that can be observed or recorded .

What is a trial in the context of probability?

In probability, a trial is one of the repeats of a probability experiment .

What is an outcome in the context of probability?

In probability, an outcome is a possible result of a trial .

Define sample space .

A sample space is the set of all possible outcomes of an experiment.

True or False? The sum of the probabilities of all outcomes in a sample space can be less than 1 .

False. The sum of the probabilities of all outcomes in a sample space is equal to 1 .

What are mutually exclusive events?

Mutually exclusive events are events that cannot both occur at the same time. For example, 'roll a 1' and 'roll an even number' are mutually exclusive events, because they cannot both occur on the same roll of a dice.

What are independent events?

Independent events are events where one occurring (or not) does not affect the probability of the other occurring.

What is relative frequency ?

Relative frequency is the experimental probability of an outcome. It is calculated by dividing the frequency of the outcome (i.e. the number of times the outcome occurs) by the number of trials .

What is conditional probability ?

Conditional probability is where the probability of an event happening can vary depending on the outcome of another event .

Define without replacement in the context of probability.

In the context of probability, without replacement is when items are not returned to the set after being selected. This changes the probabilities for subsequent selections.

True or False? In conditional probability questions, the total number of items remains constant when selecting without replacement .

False. In conditional probability questions, the total number of items changes when selecting without replacement .

What is Bayes' Theorem used for?

Bayes' Theorem is used to find conditional probabilities if you switch the order of the events. If you have a tree diagram for event B followed by event A , Bayes' Theorem lets you find the conditional probabilities needed to draw a tree diagram for event A followed by event B .

True or False? In a Venn diagram , all the circles must overlap .

False. In a Venn diagram , circles may overlap, depending on whether or not outcomes are shared between events, but they don't have to overlap. In particular, mutually exclusive events are represented by non-overlapping circles in a Venn diagram.

True or False? Independent events can be seen instantly in a Venn diagram.

False. Independent events can not be seen instantly in a Venn diagram. You need to use probabilities to deduce if two events are independent.

True or False? In a Venn diagram showing frequencies , all the frequencies shown in the diagram should add up to the total frequency .

True. In a Venn diagram showing frequencies , all the frequencies shown in the diagram should add up to the total frequency .

True or False? The events on the branches of a tree diagram must be mutually exclusive .

True. The events on the branches of a tree diagram must be mutually exclusive .

True or False? On a tree diagram , all the probabilities on a single set of branches should add up to 1 .

True. On a tree diagram , all the probabilities on a single set of branches should add up to 1 .

How do you calculate the probability of two events happening together in a tree diagram?

To find the probability that two events happen together in a tree diagram, you multiply the corresponding probabilities along their branches.

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Define the term experiment in the context of probability.
In probability, an experiment is a repeatable activity that has a result that can be observed or recorded.
What is a trial in the context of probability?
In probability, a trial is one of the repeats of a probability experiment.
What is an outcome in the context of probability?
In probability, an outcome is a possible result of a trial.
Define sample space.
A sample space is the set of all possible outcomes of an experiment.
State the equation for the theoretical probability of event $A$ occurring, in terms of $n (A)$ and $n (U)$ .
The theoretical probability of event $A$ occurring is $P (A) = \frac{n (A)}{n (U)}$
Where:
- $P (A)$ is the probability of event $A$ occurring
- $n (A)$ is the number of outcomes in the sample space that belong to event $A$
- $n (U)$ is the total number of possible outcomes in the sample space
This equation assumes that all outcomes in the sample space are equally likely.
The equation is in the exam formula booklet.
What does $A^{'}$ denote, in relation to an event $A$ ?
$A^{'}$ is the complement of event $A$ .
This can be thought of as 'not $A$ '. It is the event where $A$ doesn't happen.
State the equation that connects the probabilities $P (A)$ and $P (A^{'})$ .
The equation that connects the probabilities $P (A)$ and $P (A^{'})$ is the complementary events equation $P (A) + P (A^{'}) = 1$
Where:
- $P (A)$ is the probability of event $A$ occurring
- $P (A^{'})$ is the probability of event $A$ not occurring
This equation is in the exam formula booklet.
True or False?
The sum of the probabilities of all outcomes in a sample space can be less than 1.
False.
The sum of the probabilities of all outcomes in a sample space is equal to 1.
What is the intersection of two events?
The intersection of two events is the event where both of the events occur.
If the two events are $A$ and $B$ , then this is the event ( $A$ and $B$ ). It is denoted by $A \cap B$ .
What is $A \cup B$ the notation for?
$A \cup B$ is the notation for the union of events $A$ and $B$ .
This is the event where $A$ or $B$ (or both) occurs.
State the equation that connects the probabilities $P (A)$ , $P (B)$ , $P (A \cup B)$ and $P (A \cap B) .$
The equation that connects the probabilities $P (A)$ , $P (B)$ , $P (A \cup B)$ and $P (A \cap B)$ is the combined events equation $P (A \cup B) = P (A) + P (B) - P (A \cap B)$
Where:
- $P (A)$ is the probability of event $A$ occurring
- $P (B)$ is the probability of event $B$ occurring
- $P (A \cup B)$ is the probability of event $A$ or event $B$ occurring
- $P (A \cap B)$ is the probability of event $A$ and event $B$ occurring
This equation is in the exam formula booklet.
What are mutually exclusive events?
Mutually exclusive events are events that cannot both occur at the same time.
For example, 'roll a 1' and 'roll an even number' are mutually exclusive events, because they cannot both occur on the same roll of a dice.
What are independent events?
Independent events are events where one occurring (or not) does not affect the probability of the other occurring.
True or False?
If $A$ and $B$ are independent events, then $P (A \cap B) = P (A) + P (B)$ .
False.
If $A$ and $B$ are independent events, then $P (A \cap B) = P (A) P (B)$ .
I.e. multiply the probabilities to find the probability of both events occurring.
If $A$ and $B$ are mutually exclusive events, then $P (A \cup B) = P (A) + P (B)$ .
I.e. add the probabilities to find the probability of one or the other (or both) events occurring.
Both of those equations are in the exam formula booklet.
State the equation that connects the probabilities $P (A)$ , $P (B)$ and $P (A \cup B)$ for mutually exclusive events.
The equation that connects the probabilities $P (A)$ , $P (B)$ , $P (A \cup B)$ for mutually exclusive events is $P (A \cup B) = P (A) + P (B)$ .
Where:
- $P (A)$ is the probability of event $A$ occurring
- $P (B)$ is the probability of event $B$ occurring
- $P (A \cup B)$ is the probability of event $A$ or event $B$ occurring
This equation is in the exam formula booklet.
What is relative frequency?
Relative frequency is the experimental probability of an outcome.
It is calculated by dividing the frequency of the outcome (i.e. the number of times the outcome occurs) by the number of trials.
True or False?
For two mutually exclusive events $A$ and $B$ , the probability $P (A \cap B)$ can take on any value between 0 and 1.
False.
For two mutually exclusive events $A$ and $B$ , $P (A \cap B) = 0$ .
This is because if $A$ and $B$ are mutually exclusive, they cannot both occur.
What is conditional probability?
Conditional probability is where the probability of an event happening can vary depending on the outcome of another event.
What is $P (A | B)$ the notation for?
$P (A | B)$ is the notation for the conditional probability of event $A$ occurring, given that event $B$ has occurred.
Define without replacement in the context of probability.
In the context of probability, without replacement is when items are not returned to the set after being selected.
This changes the probabilities for subsequent selections.
True or False?
In conditional probability questions, the total number of items remains constant when selecting without replacement.
False.
In conditional probability questions, the total number of items changes when selecting without replacement.
True or False?
The equation for the conditional probability of $A$ given $B$ is $P (A | B) = \frac{P (A \cup B)}{P (A)}$ .
False.
The equation for the conditional probability of $A$ given $B$ is $P (A | B) = \frac{P (A \cap B)}{P (B)}$ .
This is given in the exam formula booklet.
True or False?
Conditional probability can be calculated using sample space diagrams.
True.
Conditional probability can be calculated using sample space diagrams.
For example, to find $P (A | B)$ using a sample space diagram:
- reduce your sample space to just include outcomes for event B,
- then find the proportion of those outcomes that also contains outcomes for event A.
What is the formula for $P (A \cap B)$ in terms of $P (A | B)$ and $P (B)$ ?
A formula for $P (A \cap B)$ can be written in the form $P (A \cap B) = P (A | B) \times P (B)$ .
This is not in the exam formula booklet.
But it can be found by rearranging $P (A | B) = \frac{P (A \cap B)}{P (B)}$ , which is in the formula booklet.
True or False?
$P (B \cap A) = P (A \cap B)$
True.
$P (B \cap A) = P (A \cap B)$
The events $B \cap A$ ( $B$ and $A$ both occur) and $A \cap B$ ( $A$ and $B$ both occur) are the same event, so their probabilities are the same.
True or False?
$P (B | A) = P (A | B)$
False.
In general $P (B | A)$ , the probability that $B$ occurs given that $A$ has occurred, is not equal to $P (A | B)$ , the probability that $A$ occurs given that $B$ has occurred.
What is the relationship between $P (A | B)$ and $P (A)$ if $A$ and $B$ are independent?
If $A$ and $B$ are independent, then $P (A | B) = P (A)$ .
True or False?
Conditional probabilities can be used to test whether two events are independent.
True.
If $A$ and $B$ are two events then they are independent if $P (A | B) = P (A) = P (A | B')$ .
I.e., if the probability of $A$ happening does not depend on whether $B$ happens or not, then the two events are independent.
What is Bayes' Theorem used for?
Bayes' Theorem is used to find conditional probabilities if you switch the order of the events.
If you have a tree diagram for event B followed by event A, Bayes' Theorem lets you find the conditional probabilities needed to draw a tree diagram for event A followed by event B.
How could you use Bayes' Theorem to calculate $P (B | A)$ ?
To calculate $P (A | B)$ :
- find $P (B)$ by multiplying along the branches and adding together the combinations that involve $B$ : $P (A) P (B | A) + P (A') P (B | A')$ ,
- find $P (A \cap B)$ by multiplying along the branches: $P (A) P (B | A)$ ,
- calculate $\frac{P (A \cap B)}{P (B)}$ .
You can get this by using Bayes' Theorem, a version of it is given in the exam formula booklet.
$P (A | B) = \frac{P (A) P (B | A)}{P (A) P (B | A) + P (A') P (B | A')}$
What is Bayes' Theorem formula to calculate $P (B_{2} | A)$ ?
$P (B_{2} | A) = \frac{P (B_{2}) P (A | B_{2})}{P (B_{1}) P (A | B_{1}) + P (B_{2}) P (A | B_{2}) + P (B_{3}) P (A | B_{3})}$ .
A version of this is given in the exam formula booklet.
True or False?
If you know $P (B)$ , $P (A | B)$ and $P (A | B')$ , then you can calculate $P (B | A)$ .
True.
If you know $P (B)$ , $P (A | B)$ and $P (A | B')$ , then you can calculate $P (B | A)$ using Bayes' Theorem.
You also need $P (B')$ but this can be found using $P (B') = 1 - P (B)$ .
What is a Venn diagram?
A Venn diagram is a way to illustrate events from an experiment, and is particularly useful when there is an overlap between possible outcomes.
What does the rectangle in a Venn diagram represent?
The rectangle in a Venn diagram represents the sample space (usually denoted by $U$ ).
True or False?
In a Venn diagram, all the circles must overlap.
False.
In a Venn diagram, circles may overlap, depending on whether or not outcomes are shared between events, but they don't have to overlap.
In particular, mutually exclusive events are represented by non-overlapping circles in a Venn diagram.
What does $A \cap B$ represent in a Venn diagram?
In a Venn diagram, $A \cap B$ represents the region where the $A$ and $B$ circles overlap.
What does $A^{'}$ represent in a Venn diagram?
$A^{'}$ represents the regions that are not in the $A$ circle in a Venn diagram.
What does $A \cup B$ represent in a Venn diagram?
In a Venn diagram, $A \cup B$ represents the regions that are in $A$ or $B$ or both.
True or False?
Independent events can be seen instantly in a Venn diagram.
False.
Independent events can not be seen instantly in a Venn diagram. You need to use probabilities to deduce if two events are independent.
True or False?
In a Venn diagram showing frequencies, all the frequencies shown in the diagram should add up to the total frequency.
True.
In a Venn diagram showing frequencies, all the frequencies shown in the diagram should add up to the total frequency.
What is a tree diagram?
A tree diagram is a way to show the outcomes of combined events.
True or False?
The events on the branches of a tree diagram must be mutually exclusive.
True.
The events on the branches of a tree diagram must be mutually exclusive.
True or False?
On a tree diagram, all the probabilities on a single set of branches should add up to 1.
True.
On a tree diagram, all the probabilities on a single set of branches should add up to 1.
How do you calculate the probability of two events happening together in a tree diagram?
To find the probability that two events happen together in a tree diagram, you multiply the corresponding probabilities along their branches.
How do you find $P (A \cup B)$ using a tree diagram?
To find $P (A \cup B)$ using a tree diagram, add together the probabilities of the combined outcomes that are part of that event:
$P (A \cup B) = P (A \cap B) + P (A \cap B^{'}) + P (A^{'} \cup B)$
Alternatively, subtract the probability of the combined outcome that isn't part of that event from 1:
$P (A \cup B) = 1 - P (A^{'} \cup B^{'})$

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