probability is a measure of the likelihood of a random phenomenon or chance behavior. probability...

Probability is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty.

Use the probability applet to simulate flipping a coin 100 times. Plot the proportion of heads against the number of flips. Repeat the simulation.

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Probability deals with experiments that yield random short-term results or outcomes, yet reveal long-term predictability.

The long-term proportion with which a certain outcome is observed is the probability of that outcome.


The Law of Large Numbers

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.


In probability, an experiment is any process that can be repeated in which the results are uncertain.

The sample space, S, of a probability experiment is the collection of all possible outcomes.

An even is any collection of outcomes from a probability experiment. An event may consist of one outcome or more than one outcome. We will denote events with one outcome, sometimes called simple events, ei. In general, events are denoted using capital letters such as E.


A probability model lists the possible outcomes of a probability experiment and each outcome’s probability. A probability model must satisfy rules 1 and 2 of the rules of probabilities.


If an event is a certainty, the probability of the event is 1.

If an event is impossible, the probability of the event is 0.

An unusual event is an event that has a low probability of occurring.


The classical method of computing probabilities requires equally likely outcomes.

An experiment is said to have equally likely outcomes when each simple event has the same probability of occurring.


The subjective probability of an outcome is a probability obtained on the basis of personal judgment.

For example, an economist predicting there is a 20% chance of recession next year would be a subjective probability.


Two events are disjoint if they have no outcomes in common. Another name for disjoint events is mutually exclusive events.


Complement of an Event

Let S denote the sample space of a probability experiment and let E denote an event. The complement of E, denoted EC, is all outcomes in the sample space S that are not outcomes in the event E.


Complement Rule

If E represents any event and EC represents the complement of E, then

P(EC) = 1 – P(E)


Two events E and F are independent if the occurrence of event E in a probability experiment does not affect the probability of event F. Two events are dependent if the occurrence of event E in a probability experiment affects the probability of event F.


Conditional Probability

The notation P(F | E) is read “the probability of event F given event E”. It is the probability of an event F given the occurrence of the event E.


A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol

nCr represents the number of combinations of n distinct objects taken r at a time, where r < n.


● A partition of the sample space S are twonon-empty sets A1 and A2 that divide up S

● In other words A1 ≠ Ø

A2 ≠ Ø

A1 ∩ A2 = Ø (there is no overlap)

A1 U A2 = S (they cover all of S)


● Let E be any event in the sample space S

● Because A1 and A2 are disjoint, E ∩ A1 andE ∩ A2 are also disjoint

● Because A1 and A2 cover all of S, E ∩ A1

and E ∩ A2 cover all of E

● This means that we have divided E into two disjoint pieces

E = (E ∩ A1) U (E ∩ A2)


● Because E ∩ A1 and E ∩ A2 are disjoint, we can use the Addition Rule

P(E) = P(E ∩ A1) + P(E ∩ A2)

● We now use the General Multiplication Rule on each of the P(E ∩ A1) and P(E ∩ A2) terms

P(E) = P(A1) • P(E | A1) + P(A2) • P(E | A2)


P(E) = P(A1) • P(E | A1) + P(A2) • P(E | A2)

● This is the Rule of Total Probability (for a partition into two sets A1 and A2)

● It is useful when we want to compute a probability (P(E)) but we know only pieces of it (such as P(E | A1))

● The Rule of Total Probability tells us how to put the probabilities together


● The general Rule of Total Probability assumes that we have a partition (the general definition) of S into n different subsets A1, A2, …, An Each subset is non-empty None of the subsets overlap S is covered completely by the union of the subsets

● This is like the partition before, just that S is broken up into many pieces, instead of just two pieces


• The Rule of Total Probability– Know P(Ai) and P(E | Ai)– Solve for P(E)

• Bayes’s Rule– Know P(E) and P(E | Ai)

– Solve for P(Ai | E)


probability is a measure of the likelihood of a random phenomenon or chance behavior. probability...

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