S-Cool Revision Summary
S-Cool Revision Summary
The number of permutations of arranging n distinct (different) objects is:
n! (n factorial)
n! = n x (n - 1) x (n - 2) x ... x 2 x 1
The number of ways of arranging n objects of which r are the same is:
In addition to this the number of ways of arranging n objects of p of one type are alike, q of a second type are alike, r of a third type are alike etc.
This is given by:
The number of permutations of r objects from n is written as npr.
Handy hint: Nearly all permutation questions involve putting things in order from a line where the order matters. For example, ABC is a different permutation to ACB.
Suppose that we wish to choose r objects from n, but the order in which the objects are arranged does not matter. Such a choice is called a combination. ABC would be the same combination as ACB as they include all the same letters.
The number of combinations of r objects from n, distinct, objects can be written in 2 ways:
The probability that an event, A, will happen is written as P(A).
The probability that the event A, does not happen is called the complement of A and is written as A'
As either A must or must not happen then,
P(A') = 1 - P(A) as probability of a certainty is equal to 1.
If A and B are two events then:
A B represents the event 'both A and B occur'
A B represents the event 'either A or B occur'
Mutually Exclusive Events
Two events are mutually exclusive if the event of one happening excludes the other from happening. In other words, they both cannot happen simultaneously.
For exclusive events A and B then:
P(A or B) = P(A) + P(B) this can be written in set notation as
P(A B) = P(A) + P(B)
This can be extended for three or more exclusive events
P(A or B or C) = P(A) + P(B) + P(C)
Handy hint: Exclusive events will involve the words 'or', 'either' or something which implies 'or'. Remember 'OR' means 'add'.
Two events are independent if the occurrence of one happening does not affect the occurrence of the other.
For independent events A and B then: P(A and B) = P(A) + P(B)
This can be written in set notation as: P(A B) = P(A) + P(B)
Again, this result can be extended for three or more events: P(A and B and C) = P(A) + P(B) + P(C)
Handy hint: Independent events will involve the words 'and', 'both' or something which implies either of these.
Remember 'and' means 'multiply'.
Most problems will involve a combination of exclusive and independent events. One of the best ways to answer these questions is to draw a tree diagram to cover all the arrangements.
The Addition Law
If two events, A and B, are not mutually exclusive then the probability that A or B will occur is given by the addition formula: P(A B) = P(A) + P(B) - P(A B)
The probability of A or B occurring is the probability of A add the probability of B minus the probability that they both occur.
If we need to find the probability of an event occurring given that another event has already occurred then we are dealing with Conditional probability.
If A and B are two events then the conditional probability that A occurs given that B already has is written as where:
If we rearrange this formula we obtain another useful result:
If the two events A and B are independent (for instance, one doesn't affect the other), then quite clearly,
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