WEEK 2
PROBABILITY CALCULUS
. What is probability?
Random experiment = an experiment with an uncertain outcome
2
Sample space (S) of a random experiment is a list of all possible outcome
Oi:S= (O1, O2, O3…) where I indicates the number of the particular possible
outcome (sample space set up correctly so experiment results in one of the
outcomes)
– e.g. experiment coin: “flip a coin” has sample space (heads, tails)
– e.g. experiment opinion “draw a random customer and record their
opinion about a service” has sample space (very bad, bad, neutral, good,
very good)
Probability of an outcome = long-run relative frequency of the outcome,
when experiment has been repeated an “infinite” number of times - P(Oi)
S = (O1, O2, …)
– 0 ≤ P(Oi) ≤ 1 for every possible outcome Oi
– ΣP(Oi) = 1 (sum of probabilities of all possible outcomes)
If all N possible outcomes are equally likely then each possible outcome in
the long run has a relative frequency 1/N
Event = a collection of one or more outcomes in the sample space
Probability of an event = the sum of the probabilities of the relevant
outcomes
. Simple, joint and conditional probability
– Simple (or marginal) probability = when A is an event, then P(A) -
1 involves one event
– Joint (or simultaneous) probability = if events are A and B, the P(A and
B) - it is the probability of the joint occurrence of two events
– A and B must occur simultaneously
, – Conditional probability = the probability of one event, given the
occurrence of another event
– P (A|B) - prob. of event A given B occurs
– Also written as P(A and B)/P(B)
. Rules, exclusive and independent
Complement = all outcomes that are not in the event
= complement rule
– Because will either be A or its complement Ac
Mutually exclusive = when two events cant occur at the same time
– P (A and B) = 0
Addition rule:
P (A or B) = P(A) + P(B) - P(A and B)
– If A and B are mutually exclusive then: P (A or B) = P(A) + P(B)
Independent events = the fact that B occurs does not affect the probability
of A occurring
– P(A) = P(A|B) = P(A|Bc) -> prob. of A is the same regardless if B occurs
or not
– Two events are independent if the occurrence of one of the events does
not change the probability of getting the other
Multiplication rule
= P(A) x P(B) because P(B|A) = P(B)
. Random variables and probability distributions
A function is called random variable because the outcome is random
PROBABILITY CALCULUS
. What is probability?
Random experiment = an experiment with an uncertain outcome
2
Sample space (S) of a random experiment is a list of all possible outcome
Oi:S= (O1, O2, O3…) where I indicates the number of the particular possible
outcome (sample space set up correctly so experiment results in one of the
outcomes)
– e.g. experiment coin: “flip a coin” has sample space (heads, tails)
– e.g. experiment opinion “draw a random customer and record their
opinion about a service” has sample space (very bad, bad, neutral, good,
very good)
Probability of an outcome = long-run relative frequency of the outcome,
when experiment has been repeated an “infinite” number of times - P(Oi)
S = (O1, O2, …)
– 0 ≤ P(Oi) ≤ 1 for every possible outcome Oi
– ΣP(Oi) = 1 (sum of probabilities of all possible outcomes)
If all N possible outcomes are equally likely then each possible outcome in
the long run has a relative frequency 1/N
Event = a collection of one or more outcomes in the sample space
Probability of an event = the sum of the probabilities of the relevant
outcomes
. Simple, joint and conditional probability
– Simple (or marginal) probability = when A is an event, then P(A) -
1 involves one event
– Joint (or simultaneous) probability = if events are A and B, the P(A and
B) - it is the probability of the joint occurrence of two events
– A and B must occur simultaneously
, – Conditional probability = the probability of one event, given the
occurrence of another event
– P (A|B) - prob. of event A given B occurs
– Also written as P(A and B)/P(B)
. Rules, exclusive and independent
Complement = all outcomes that are not in the event
= complement rule
– Because will either be A or its complement Ac
Mutually exclusive = when two events cant occur at the same time
– P (A and B) = 0
Addition rule:
P (A or B) = P(A) + P(B) - P(A and B)
– If A and B are mutually exclusive then: P (A or B) = P(A) + P(B)
Independent events = the fact that B occurs does not affect the probability
of A occurring
– P(A) = P(A|B) = P(A|Bc) -> prob. of A is the same regardless if B occurs
or not
– Two events are independent if the occurrence of one of the events does
not change the probability of getting the other
Multiplication rule
= P(A) x P(B) because P(B|A) = P(B)
. Random variables and probability distributions
A function is called random variable because the outcome is random
