Bilingual Program MATHS
UNIT 11 – PROBABILITY
Experiment: an action where the result is uncertain. Ex: Tossing a coin, throwing dice, picking a ball from a bag…..
Sample Space: all the possible outcomes of an experiment.
Sample Point: just one of the possible outcomes.
Event: An event is any collection of outcomes of an experiment. Formally, any subset of the sample space is an event.
Any event which consists of a single outcome in the sample space is called an elementary or simple event. Events which consist of more than one outcome are called compound events.
Sure or certain event: an event that is certain to occur. It is S, the sample space.
Impossible event: It is the event containing no outcomes. It is denoted by Ø. Mutually Exclusive Events: These are events that cannot occur at the same time. In other words, if there is no element that is in both A and B. The intersection of A and B is Ø. Aces and Kings are Mutually Exclusive
Hearts and Kings are not Mutually Exclusive
Two events are called not mutually
exclusive if they have at least one outcome common between them.
Operations on events: If A and B are two events in the sample space S, then: Union - The event (A
B) occurs if A or B or both A and B occur.
Intersection - The event (A Complement of A
A and
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__
B) occurs only if both A and B occur.
__
( A ) - This event occurs if and only if A does not occur.
A are complementary events.
Bilingual Program MATHS Independent and dependent events:
Independent Events: These are two or more events for which the outcome of one does not affect the other. Ex: Each toss of a fair coin is an independent event.
Dependent Events: These are events that are dependent on what occurred previously. Ex: If two cards are drawn from a deck of fifty-two cards, the likelihood of the second card being an ace is dependent on the outcome of the first four cards. After taking one card from the deck there are less cards available, so the probabilities change! Don’t forget! Replacement: When you put each card back after drawing it the chances don't change, as the events are independent. Without Replacement: The chances will change, and the events are dependent.
Probability: A probability provides description of the likely particular event.
a quantitative occurrence of a
Probability goes from 0 (impossible) to 1 (certain). It is often shown as a decimal, fraction or percentage.
If all outcomes in an experiment are equally likely, the probability of an event A is calculated using the following formula known as: LAPLACE’S RULE
p( A)
number of favorable choices for the event A Total number of possible outcomes
Example: what is the probability of getting a "Head" when tossing a coin? Number of ways it can happen: 1 (Head) Total number of outcomes: 2 (Head and Tail) 1 So the probability =
= 0.5
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Bilingual Program MATHS Probability Axioms: 1.
The probability is positive and less than or equal to 1.
2.
The probability of the sure event is 1.
3.
If A and B are mutually exclusive, then:
p (S )
p( A
0
p( A) 1
1
B)
p ( A)
p( B)
Probability Properties 1 The probability of the complementary event is: p( A)
1
2 The probability of an impossible event is zero. p (Ø.)
0
p( A)
3 The probability of the union of two events is the sum of their probabilities minus the probability of their intersection. p ( A
B)
p ( A)
p( B)
p( A
B) )
4 The sum of the probabilities of all possible outcomes equals 1.
5 If an event is a subset of another event, its probability is les s than or equal to it.
If A
B, p ( A)
p( B)
6 Two events A and B are independent
7 A tree diagram is useful for displaying all outcomes for a “multistage” experiment and determining their probabilities.
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p( A
B.)
p ( A) p ( B )
Bilingual Program MATHS Conditional Probability We have already defined dependent and independent events and seen how probability of one event relates to the probability of the other event. Having those concepts in mind, we can now look at conditional probability.
Conditional probability is denoted by the following:
p A
B
It is read as the probability that A occurs given that B has already occurred.
Summary of probabilities Event
A
not A
A or B
A and B
A given B
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Probability