Probability
• Two coins are tossed. What is the probability both come up heads?
• Two six-sided dice are rolled. What is the probability they are both an even
number, but not equal?
These are two examples of probability experiments .
In a probability experiment, we cannot predict individual outcomes with certainty,
but we may know the long-run distribution of the outcomes.
One interpretation of probability:
The probability of an outcome is the proportion of times that outcome would occur
in a very long (inﬁnite) series of trials.
Sample Spaces, Events, and Simple Events
The sample space of an experiment is the set of all possible outcomes.
Simple events (also known as sample points) are the most basic outcome of
the experiment.
Examples:
• Consider rolling an ordinary six-sided die.
• Consider drawing a card from an ordinary 52 card deck.
An event is a subset of the sample space. For example, consider the following 3
events involving the roll of a six-sided die:
• Let event E represent rolling a one, two, or three:
• Let event F represent rolling a two, three or four:
• Let event G represent rolling a one or a ﬁve:
To visualize the various relationships between events, it often helps to illustrate
the events with a Venn diagram .
If all of the simple events are equally likely, then the probability of an event A is: Basic Rules of Probability
The Intersection of Events
The intersection of events A and B is the event that both A and B occur.
Mutually Exclusive Events
Events are mutually exclusive(sometimes called disjoint) if they have no events
in common.
The Union of Events
The union of events A and B is the event that either A or B or both occurs.
Complementary Events
The complement of an event A, denoted by A , is the event that A does not
occur.
Example. To illustrate the above concepts, let’s return to the die example. Recall:
• Event E represents rolling a one, two, or three: E = {1,2,3}.
• Event F represents rolling a two, three, or four: F = {2,3,4}.
• Event G represents rolling a one or a ﬁve: G = {1,5}. Conditional Probability
Consider the following two problems:
• A card is drawn from a well-shuﬄed 52 card deck. What is the probability
it is a king?
• A card is drawn from a well-shuﬄed 52 card deck. You catch a glimpse of
the card and see only that it is a face card. What is the probability it is a
king, given it is a face card?
The conditional probability of event A, given that B has occurred, is:
Independent Events
If the occurrence or nonoccurrence of A does not change the probability of B,we
say that A and B are independent. More for

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