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Probability Distributions

# Probability Distributions - Probability Distributions...

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Probability Distributions Probability and Independence Probability For an experiment we define an event to be any collection of possible outcomes . A simple event is an event that consists of exactly one outcome. or : means the union i.e. either can occur and : means intersection i.e. both must occur Two events are mutually exclusive if they cannot occur simultaneously . For a Venn diagram, we can tell that two events are mutually exclusive if their regions do not intersect We define Probability of an event E to be to be number of simple events within E P(E) = total number of possible outcomes We have the following: 1. P(E) is always between 0 and 1 . 2. The sum of the probabilities of all simple events must be 1 . 3. P(E) + P(not E) = 1 4. If E and F are mutually exclusive then P(E or F) = P(E) + P(F) The Difference Between And and Or If E and F are events then we use the terminology E and F to mean all outcomes that belong to both E and F We use the terminology

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E Or F to mean all outcomes that belong to either E or F. Example Below is an example of two sets, A and B , graphed in a Venn diagram. The green area represents A and B while all areas with color represent A or B Example Our Women's Volleyball team is recruiting for new members. Suppose that a person inquires about the team. Let E be the event that the person is female Let F be the event that the person is a student then E And F represents the qualifications for being a member of the team. Note that E Or F is not enough. We define Definition of Conditional Probability P(E and F)
P(E|F) = P(F) We read the left hand side as "The probability of event E given event F " We call two events independent if For Independent Events P(E|F) = P(E) Equivalently, we can say that E and F are independent if For Independent Events P(E and F) = P(E)P(F) Example Consider rolling two dice. Let E be the event that the first die is a 3 . F be the event that the sum of the dice is an 8 . Then E and F means that we rolled a three and then we rolled a 5 This probability is 1/36 since there are 36 possible pairs and only one of them is (3,5) We have P(E) = 1/6 And note that (2,6),(3,5),(4,4),(5,3), and (6,2) give F Hence P(F) = 5/36 We have P(E) P(F) = (1/6) (5/36) which is not 1/36 . We can conclude that E and F are not independent. Exercise

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Test the following two events for independence: E the event that the first die is a 1 . F the event that the sum is a 7 . Hold your mouse over the yellow rectangle for the answer. A Counting Rule For two events, E and F , we always have P(E or F) = P(E) + P(F) - P(E and F) Example Find the probability of selecting either a heart or a face card from a 52 card deck. Solution We let E = the event that a heart is selected F = the event that a face card is selected then P(E) = 1/4 and P(F) = 3/13 (Jack, Queen, or King out of 13 choices) P(E and F) = 3/52 The formula gives P(E or F) = 1/4 + 3/13 - 3/52 = 22/52 = 42% Trees and Counting Using Trees We have seen that probability is defined by
Number in E P(E) = Number in the Sample Space Although this formula appears simple, counting the number in each can prove to be a challenge.

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Probability Distributions - Probability Distributions...

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