PROBABILITY
REVISION
Outcome: a single observation of an uncertain or random process (called
an experiment). For example, when you accidentally drop a book, it might fall on
its cover, on its back or on its side. Each of these options is a possible outcome.
Sample space of an experiment: the set of all possible outcomes of the
experiment. For example, the sample space when you roll a single 6-sided die is
the set {1; 2; 3; 4; 5; 6}. For a given experiment, there is exactly one sample space.
The sample space is denoted by the letter 𝑆.
Event: a set of outcomes of an experiment. For example, if you have a standard
deck of 52 cards, an event may be picking a spade card or a king card.
Probability of an event: a real number between and inclusive of 00 and 11 that
describes how likely it is that the event will occur. A probability of 00means the
outcome of the experiment will never be in the event set. A probability
of 11 means the outcome of the experiment will always be in the event set. When
all possible outcomes of an experiment have equal chance of occurring, the
probability of an event is the number of outcomes in the event set as a fraction
of the number of outcomes in the sample space. To calculate a probability, you
divide the number of favourable outcomes by the total number of possible
outcomes.
Relative frequency of an event: the number of times that the event occurs
during experimental trials, divided by the total number of trials conducted. For
example, if we flip a coin 10 times and it landed on heads 3 times, then the
*
relative frequency of the heads event is = 0,3
+,
Union of events: the set of all outcomes that occur in at least one of the events.
For 2 events called A and B, we write the union as “A or B”. Another way of
writing the union is using set notation: 𝐴 ∪ 𝐵. For example, if A is all the countries
in Africa and B is all the countries in Europe, A or B is all the countries in Africa
and Europe.
Intersection of events: the set of all outcomes that occur in all of the events.
For 2 events called A and B, we write the intersection as “A and B”. Another way
of writing the intersection is using set notation: 𝐴 ∩ 𝐵. For example, if A is soccer
players and B is cricket players, A and B refers to those who play both soccer
and cricket.
Mutually exclusive events: events with no outcomes in common, that is (A and B)
is an empty set. Mutually exclusive events can never occur simultaneously. For
example the event that a number is even and the event that the same number is
odd are mutually exclusive, since a number can never be both even and odd.
SASHTI NOTES 1
, PROBABILITY
Complementary events: two mutually exclusive events that together contain all
the outcomes in the sample space. For an event called A, we write the
complement as “not A”. Another way of writing the complement is as A′.
Dependent and independent events:
Two events, A and B, are independent if the outcome of the first event
does not influence the outcome of the second event. For example, if you
flip a coin and it lands on tails and flip it again and it lands on heads,
neither outcome influences the other.
Two events, C and D, are dependent if the outcome of one event
influences the outcome of the other. For example, if your lunchbox
contains 3 sandwiches and 2 apples, when you eat one of the items, this
reduces the number of choices you have when deciding to eat a second
item.
IDENTITIES
The addition rule (also called the sum rule) for any 2 events, A and B is
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
This rule relates the probabilities of 2 events with the probabilities of their union
and intersection.
The addition rule for 2 mutually exclusive events is
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
This rule is a special case of the previous rule. Because the events are mutually
exclusive, 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0
The complementary rule is 𝑃(𝑛𝑜𝑡 𝐴) = 1 − 𝑃(𝐴)
This rule is a special case of the previous rule. Since A and (not A) are
complementary, 𝑃(𝐴 𝑜𝑟 (𝑛𝑜𝑡 𝐴)) = 1
The product rule for independent events A and B is:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
If two events A and B are dependent then:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) ≠ 𝑃(𝐴) × 𝑃(𝐵)
SASHTI NOTES 2
REVISION
Outcome: a single observation of an uncertain or random process (called
an experiment). For example, when you accidentally drop a book, it might fall on
its cover, on its back or on its side. Each of these options is a possible outcome.
Sample space of an experiment: the set of all possible outcomes of the
experiment. For example, the sample space when you roll a single 6-sided die is
the set {1; 2; 3; 4; 5; 6}. For a given experiment, there is exactly one sample space.
The sample space is denoted by the letter 𝑆.
Event: a set of outcomes of an experiment. For example, if you have a standard
deck of 52 cards, an event may be picking a spade card or a king card.
Probability of an event: a real number between and inclusive of 00 and 11 that
describes how likely it is that the event will occur. A probability of 00means the
outcome of the experiment will never be in the event set. A probability
of 11 means the outcome of the experiment will always be in the event set. When
all possible outcomes of an experiment have equal chance of occurring, the
probability of an event is the number of outcomes in the event set as a fraction
of the number of outcomes in the sample space. To calculate a probability, you
divide the number of favourable outcomes by the total number of possible
outcomes.
Relative frequency of an event: the number of times that the event occurs
during experimental trials, divided by the total number of trials conducted. For
example, if we flip a coin 10 times and it landed on heads 3 times, then the
*
relative frequency of the heads event is = 0,3
+,
Union of events: the set of all outcomes that occur in at least one of the events.
For 2 events called A and B, we write the union as “A or B”. Another way of
writing the union is using set notation: 𝐴 ∪ 𝐵. For example, if A is all the countries
in Africa and B is all the countries in Europe, A or B is all the countries in Africa
and Europe.
Intersection of events: the set of all outcomes that occur in all of the events.
For 2 events called A and B, we write the intersection as “A and B”. Another way
of writing the intersection is using set notation: 𝐴 ∩ 𝐵. For example, if A is soccer
players and B is cricket players, A and B refers to those who play both soccer
and cricket.
Mutually exclusive events: events with no outcomes in common, that is (A and B)
is an empty set. Mutually exclusive events can never occur simultaneously. For
example the event that a number is even and the event that the same number is
odd are mutually exclusive, since a number can never be both even and odd.
SASHTI NOTES 1
, PROBABILITY
Complementary events: two mutually exclusive events that together contain all
the outcomes in the sample space. For an event called A, we write the
complement as “not A”. Another way of writing the complement is as A′.
Dependent and independent events:
Two events, A and B, are independent if the outcome of the first event
does not influence the outcome of the second event. For example, if you
flip a coin and it lands on tails and flip it again and it lands on heads,
neither outcome influences the other.
Two events, C and D, are dependent if the outcome of one event
influences the outcome of the other. For example, if your lunchbox
contains 3 sandwiches and 2 apples, when you eat one of the items, this
reduces the number of choices you have when deciding to eat a second
item.
IDENTITIES
The addition rule (also called the sum rule) for any 2 events, A and B is
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 𝑎𝑛𝑑 𝐵)
This rule relates the probabilities of 2 events with the probabilities of their union
and intersection.
The addition rule for 2 mutually exclusive events is
𝑃(𝐴 𝑜𝑟 𝐵) = 𝑃(𝐴) + 𝑃(𝐵)
This rule is a special case of the previous rule. Because the events are mutually
exclusive, 𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 0
The complementary rule is 𝑃(𝑛𝑜𝑡 𝐴) = 1 − 𝑃(𝐴)
This rule is a special case of the previous rule. Since A and (not A) are
complementary, 𝑃(𝐴 𝑜𝑟 (𝑛𝑜𝑡 𝐴)) = 1
The product rule for independent events A and B is:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
If two events A and B are dependent then:
𝑃(𝐴 𝑎𝑛𝑑 𝐵) ≠ 𝑃(𝐴) × 𝑃(𝐵)
SASHTI NOTES 2
