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Probability and discrete random variables (Probability (Strategy (3 For…
Probability and discrete random variables
Probability
A random experiment is any repeatable action with a collection of clearly defined outcomes that cannot be predicted with certainity
The sample space for an experiment is the collection of all possible outcomes of the experiment
The total probability associated with a sample space is 1
The probability of an event happening is written 'P(event)'
If A and B are mutually exclusive, P(A or B) = P(A) + P(B)
P(A') = 1-P(A)
Complementary events are mutually exclsive and exhaustive- no other outcome exists
For a sample space with N equally likely outcomes, the probability of any one occuring is 1/N
If event A occurs in n(A) of the equally probable outcomes, the probability of A is given by P(A) = n(A)/N
Two events, A and B, are independent if the fact that A has occured does not affect the probability of B occuring
If A and B are independent, P(A and B) = P(A) X P(B)
P(A or B) = P(AuB)
P(A and B) = P(AnB)
The u and n symbols are called the union and intersection symbols respectively
Strategy
3 For unknown probabilities, consider using the 'probabilities total 1' result
1 Identify mutually exclusive events and use the addition rule
2 Identify independent events and use the multiplication rule
Binomial distribution
Conditions for a binomial probability
Two pssible outcomes in each trial
Fixed number of trials
Independent trials
Identical trials (p is the same for each trail)
P(X=x) = nCx p^X(1-p)^(n-x)
where n is the number of trials and p is the probability of success in any given trial
Written as X~B(n,p)
If X can only take integer values, P(X<x) = P(X≤x-1) and P(X>x) = 1 - P(X≤x)
Strategy
1 Check the conditions for a binomial distribution are met. List any assumptions
2 Identify the random variable and corresponding values of n and p
3 Calculate probablities using the addition and multiplication rules if necessary