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6.1
Random – Phenomena are called random if there are uncertain individual outcomes but a regular
distribution of outcomes can be seen in very many repetitions.
Probability or long-term relative frequency – In random phenomena, the probability of each of the
outcomes is the proportion of times the outcome would take place in very many repetitions.
Things to Remember:
1) The many trials in a long series have to be independent. This means that the result of one trial does
not affect the result of another trial.
2) Real-world probabilities can only be estimated when you observe very many trials.
3) Computer simulations can be very helpful because they provide a way to have very many trials.
Various things in which probability is used:
- games of chance (casinos and lotteries)
- astronomy
- surveying
- human life span
- weight and length of skulls, cockroaches, leaves
- traffic flow through highways system
- traffic flow through a computer processer
- genetic makeup of individuals in a population
- subatomic particles’ energy states
- rumor or epidemic spreading
- return rate on risky investments
- STATISTICS
6.2
A Model of Randomness includes:
- All of the possible outcomes listed
• sample space S – the set of all of the possible outcomes of a random phenomenon
- Examples of sample spaces:
• S = {heads, tails}
• S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
- Each of these outcomes has a probability listed with it
• event – a subset of the sample space, or an outcome or set of outcomes of a random phenomenon
- When assigning probabilities to events the probability must be between 0 and 1. A probability
of 0 means that the event never happens, a probability of 1 means that the event always
happens, and a probability of 0.5 means that the event occurs in half of the trials.
• When all of the probabilities of all of the possible outcomes are added together, the sum must be 1.
• To find the probability of an event not occurring, you subtract the probability that the event does
occur from 1. If you add the probability that an event does occur, with the probability that an event
does not occur, you will get 1.
• If there are no outcomes in common between two events, the probability of one or the other occurring
is the sum of the two probabilities of the two outcomes.
Probability Rules
1) The probability P(A) of any event is greater than or equal to 0, and less than or equal to 1.
2) The probability of the sample space is equal to 1.
3) The probability that an event A does not occur is the complement of A which is written: Ac. The
complement rule states that:
P(Ac) = 1 – P(A)
4) Events A and B can be called disjoint if none of their outcomes are in common and so they can
never occur at the same time. If two events are disjoint then the addition rule for disjoint events says
that:
P(A or B) = P(A) + P(B)
5) Events A and B can be called independent if knowing that one of the events takes place does not
change the probability of the other event taking place. If in fact the two events A and B are
independent, the multiplication rule for independent events says:
P (A and B) = P(A)P(B)
Venn diagrams, like the ones above, can be used to explain the rules of probability in picture form.
Probabilities in a Finite Sample Space
1) Assign probabilities to each of the individual outcomes. Remember that each of these probabilities
must be between 0 and 1, and the probabilities of all of the outcomes must have a sum of 1.
2) Add up the probabilities of the outcomes that make up an event, and then you will have the
probability of that event.
If all of the outcomes are equally likely, you can assign probabilities to events in this way:
1) If k possible outcomes of a random phenomenon are all equally likely, then each of the outcomes
has a probability of 1/k.
2) Then, the probability of any event is:
P (A) = count of outcomes in A/count of outcomes in S
or
P (A) = count of outcomes in A/k
* Disjoint events are not independent.*
*If events A and B are independent, their complements are also independent. This means that Ac and
Bc are also independent. Also, Ac is independent of B.*
The probability rule of multiplication can also be extended for circumstances with more than two
events.
6.3
Union – This is the event that at least one of the collection of events occurs.
• For events A and B, the union can be described in the way that A or B or both A and B occur.
Addition Rule for Disjoint Events
• If events A, B, and C have no outcomes in common and thus are disjoint, then:
P (one or more of A, B, C) = P(A) + P(B) + P(C)
General Addition Rule for Unions of Two Events
• For events A and B: P (A or B) = P(A) + P(B) – P (A and B)
• If the events A and B are disjoint then the P (A and B) will be 0 because it will be an empty
event because it will have no outcomes in it.
Conditional Probability – Conditional probability gives the probability of one events given that we
know the probability of another event.
The form which a conditional probability is said in is: The probability of ___________ given that
_______________. Conditional probability is also expressed as P(B│A), where the bar represents the
words “given that”.
To find a conditional probability in a two-way table, it is easiest to use your hands to cover up all of the
rows or columns that are not necessary. Cover up all of the rows and columns that do not have to do
with the words after the words “given that”. Then, within that specific row and column, find the
probability of the event that is given before the words, “given that”.
Multiplication Rule – The probability that events A and B take place together is: P (A and B) =
P(A)P(B│A)
For three events, the multiplication rule is: P (A and B and C) = P(A) P(B│A) P(C│A and B)
Definition of Conditional Probability – The conditional probability of B given A (when P(A) is
greater than 0), is: P(B│A) = P(A and B)│P(A)
Intersection – The event that all of the events occur in any number of events is the intersection.
To solve complex problems that give many different probabilities of different events, we can use tree
diagrams. Both the addition and multiplication probability rules can be used in tree diagrams to find
specific probabilities.
This is an example of a tree diagram:
QuickTime™ and a
BMP decompressor
are needed to see this picture.
With a tree diagram, if you are trying to find the probability of getting to the end of the segment, all
you have to do is multiply all of the probabilities of each of the segments leading up to the end of the
segment.
Also with a tree diagram, if you want to find the probability of any of the events, you have to add the
probabilities of all of the branches that are involved with that event to get the probability of an event.
Bayes’s Rule
With events A and B, that do not have probabilities of 0 or 1, Bayes’s Rule is this:
P (A│B) = P(B│A)P(A)│P(B│A)P(A) + P(B│Ac)P(Ac)
*It is not necessary to memorize this formula. It is much easier to just work through a problem using
tree diagrams, etc. to get the probability of event A given event B.*
Independent Events – If two events A and B have probabilities that are positive, they are
independent if:
P(B│A) = P(B)
This means that no additional information is given by knowing the probability of A, so knowing the
probability of A does not affect the probability of B, so the two probabilities are independent of each
other.