Everything Maths, Grade 10. Chapter 10: Probability.

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14. Probability
Grade 10

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Terminology
● An experiment refers to an uncertain process.
● An outcome of an experiment is a single result of that experiment.
● The sample space of an experiment is the set of all possible outcomes of that experiment.
The sample space is denoted with the symbol S and the size of the sample space (the total
number of possible outcomes) is denoted with n(S)
● An event is a specific set of outcomes of an experiment that you are interested in. An event is
denoted with the letter E and the number of outcomes in the event with n(E).

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Theoretical probability
● A probability is a real number between 0 and 1 that describes how likely it is that an
event will occur. A probability can also be written as a percentage or as a fraction.
● A probability of 0 means that an event will never occur and a probability of 1 means that
an event will always occur. A probability of 0,5 means that an event will occur half the
time.
● The probability of an event is the ratio between the number of outcomes in the event set
and the number of possible outcomes in the sample space:
Relative frequency
● The relative frequency of an event is defined as the number of times that the event occurs
during experimental trials, divided by the total number of trials conducted.

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Venn diagrams
A Venn diagram is a graphical way of representing the relationship between sets.In each Venn
diagram a set is represented by a closed curve. The region inside the curve represents the
elements that belong to the set, while the region outside the curve represents the elements
that are excluded from the set.

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Union and intersection
● The union of two sets is a new set that contains all of the elements that are in at least one
of the two sets. The union is written as A B.∪
● The intersection of two sets is a new set that contains all of the elements that are in both
sets. The intersection is written as A ∩ B.
● Union and intersection: P (A B) = P (A) + P (B) − P (A ∩ B).∪

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Probability identities
● The probability of sample space: P(S) = 1.
● P (A B) = P (A) + P (B) − P (A ∩ B).∪

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Mutually exclusive events
Mutually exclusive events are two events that cannot occur at the same time. Whenever an
outcome of an experiment is in the first event, it can not also be in the second event.

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Complementary events
● The complement of a set, A, is a different set that contains all of the elements that are not in
A. We write the complement of A as A', or sometimes as “not (A)”.
● Complementary events are mutually exclusive: A ∩ A' = ∅
● Complementary events cover the sample space: A A' = S∪
● Probabilities of complementary events sum to 1: P (A) + P (A' ) = P (A A' ) = P (S) = 1∪

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