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Finite mathematics topic summary: probability 
Sample Space and Events
An experiment is an occurrence whose result, or outcome is uncertain. The set of all possible outcomes is called the sample space for the experiment. Given a sample space S, an event E is a subset of S. The outcomes in E are called the favorable outcomes. We say that E occurs in a particular experiment if the outcome of that experiment is one of the elements of E, that is, if the outcome of the experiment is favorable. OnLine Tutorial Beginning With This Topic 
Example
1. Experiment: Cast a die and observe the number facing up.
2. Here is an experiment that simulates tossing three fair distinguishable coins. To run the experiment, press the "Toss Coins" button and record the occurrences of heads and tails. The sample space is the set of eight possible outcomes:
Let E be the event that heads comes up at least twice.


Combining Events
If E and F are events in an experiment, then: E' is the event that E does not occur. E F is the event that either E occurs or F occurs (or both). E F is the event that both E and F occur. E and F are said to be disjoint or mutually exclusive if (E F) is empty. 
Example
Let S be the sample space for the coin tossing experiment in the box above, so that S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}. Let E be the event that heads comes up at least twice; E = {HHH, HHT, HTH, THH}, and let F be the event that tails comes up at least once;F = {HHT, HTH, HTT, THH, THT, TTH, TTT}. Then: E' = {HTT, THT, TTH, TTT}
E and F are not mutually exclusive, since E F ≠ 

Relative Frequency
If an experiment is performed N times, and the event E occurs fr(E) times, then the ratio
The number fr(E) is called the frequency of E. N, the number of times that the experiment is performed, is called the number of trials or the sample size. If E consists of a single outcome s, then we refer to P(E) as the relative frequency of the outcome s, and write P(s). OnLine Tutorial on Relative Frequency 
Example
In the above experiment (toss three coins) take E to be the event that heads comes up at least twice. To compute its relative frequency, let us use the simulated experiment below. Every time you press "Toss Coins" the web page will compute both fr(E) and P(E).


Some Properties of Relative Frequency
Let S = {s_{1}, s_{2}, ... , s_{n}} be a sample space and let P(s_{i}) be the relative frequency of the event {s_{i}}. Then
(a) 0 ≤ P(s_{i}) ≤ 1
In words: (a) The relative frequency of each outcome is a number between 0 and 1.


Modeled Probability
The modeled probability, P(E), of an event E is a mathematical model whose purpose is to predict the relative frequency based on the nature of the experiment rather than through actual experimentation. The relative frequency should approach the modeled probability as the number of trials gets larger and larger. Notes 1. We sometimes write P(E) for both relative frequency and modeled probability. Which one we are referring to should always be clear from the context.

Example
In the above experiment, there are eight outcomes in S, and half of them are in E. Therefore, we expect E to occur half the time. In other words, the modeled probability of E is 0.5. If you "toss the coins" in the simulated experiment a large number of times, the relative frequency should approach the modeled probability of .5. In the following simulation, you can toss the coins 50 times with each click on the button.


Modeled Probability for Equally Likely Outcomes
In an experiment in which all outcomes are equally likely, we model the probability of an event E by

Example
In the above experiment (toss three coins) there are eight equally likely outcomes in S, and half of them are in E (the event that heads comes up at least twice). Therefore,


Probability Distributions
Relative frequency and modeled probability have in common the idea of a probability distribution: A finite sample space is just a finite set S. A probability distribution is an assignment of a number P(s_{i}) to each outcome s_{i} in a sample space S ={s_{1}, s_{2}, ... , s_{n}} so that
(a) 0 ≤ P(s_{i}) ≤ 1
P(s_{i}) is called the probability of s_{i}. Given a probability distribution, we obtain the probability of an event E by adding up the probabilities of the outcomes in E. If P(E) = 0, we call E an impossible event. The event is always impossible, since something must happen. Notes 1. All the above properties apply to both relative frequency and modeled probability. Thus, when we speak only of "probability," we could mean either, depending on the context. OnLine Tutorial on Probability Distributions 
Example
1. All the examples of relative frequency and modeled probability above give examples of probability distributions. 2. Let us take S = {H, T} and make the assignments P(H) = 0.2, P(T) = 0.8. Since these numbers are between 0 and 1, and add to 1, they specify a probability distribution. 3. With S = (H, T} again, we could also take P(H) = 1, P(T) = 0, so that T is an impossible event. 4. The following table gives a probability distribution for the sample space S = {1, 2, 3, 4, 5, 6}.
It follows that
5. Suppose we toss three coins as above, but this time, we only look at the number of heads that come up. In other words, S = {0, 1, 2, 3}. The (modeled) probability distribution is given by counting the number of combinations that give 0, 1, 2, and 3 heads:
The following simulation computes the relative frequency distribution of the above experiment. You will find that, after many coin tosses, these relative frequencies converge to the modeled probabilities shown above.


Addition Principle
Mutually Exclusive Events
General Addition Principle

Example
Let S be the sample space for the cointossing experiment above; S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Let E be the event that heads comes up at exactly once;
Then E and F are mutually exclusive, and
Now let E be as above, and let F be the event that heads comes up at most once:


Further Properties of Probability
The following are true for any sample space S and any event E.

Example
Continuing with S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Let E be the event that heads comes up at exactly once;


Conditional Probability
If E and F are two events, then the conditional probability, P(E  F), is the probability that E occurs, given that F occurs, and is defined by
We can rewrite this formula in a form known as the multiplication principle:
Conditional Estimated Probability
Conditional Probability for Equally Likely Outcomes
OnLine Tutorial Beginning With This Topic 
Example
Let S be the original sample space for the experiment above; S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Let E be the event that heads comes up at exactly once;
Then the probability that heads comes up exactly once, given that the first comes up heads is
Since the outcomes in this experiment are equally likely, we could also use the formula


Independent Events
The events E and F are independent if
If two events E and F are not independent, then they are dependent. Given any number of mutually independent events (that is, each one of them is independent of the intersection of any combination of the others), the probability of their intersection is the product of the probabilities of the individual events. 
Example
As in the example immediately above, let S be the original sample space for the experiment above; S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT} Let E be the event that heads comes up at exactly once;
To test these two events for independence, we check the formula
E F = {HTT}, so that P(E F) = 1/8.


Bayes' Theorem
The short form of Bayes' Theorem states that if E and F are events, then
We can often compute P(F  E) by instead constructing a probability tree. (To see how, go to the tutorial by following the link below.) An expanded form of Bayes' Theorem states that if E is an event, and if F_{1}, F_{2}, and F_{3} are a partition of the sample space S, then
A similar formula works for a partition of S into four or more events. OnLine Tutorial Beginning With This Topic 
Example
If P(E  F) = 0.95 P(E  F') = 0.15 P(F) = 0.1 P(F') = 0.9, then
This example comes from a scenario discussed in the tutorial (link on the adjacent box). 