Probability and Independence
- Page ID
- 64202
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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 \[ P(E) = \frac{ \text{number of simple events within E}}{\text{total number of possible outcomes}}\] We have the following:
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 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
We read the left hand side as "The probability of event E given event F" We call two events independent if
Equivalently, we can say that E and F are independent if
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 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.
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%
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