Events are often labeled with capital letters, such as A and B.

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1 Probability Sample Space (S) -The set of all possible outcomes. We can represent the sample space using set notation, a tree diagram, or a Venn Diagram. Event (A) A specific outcome or group of outcomes. A subset of the set of all possible outcomes (S). Events are often labeled with capital letters, such as A and B.

2 Example: List the sample space for each activity or experiment. a) In a car or truck engine, a greater number of cylinders generally increases engine smoothness and power. [Duffy] Use set notation to list the sample space for the normal number of cylinders in a car or truck engine.

3 Example: List the sample space for each activity or experiment. b) In an inline engine, the cylinders are lined up in a single row. In a V-type engine, the two banks of cylinders lie at an angel to each other. A slant engine has only one bank of cylinders; this bank leans to one side. Cylinders of an opposed engine lie flat on either side of the crankshaft... Four-cylinder engines usually have inline, slant, or opposed cylinder arrangements. Six-cylinder engines can have inline, slant, or V-type arrangements. Eight-cylinder engines are commonly V-type engines. [Duffy] Use a Venn-Diagram to summarize the sample space..

4 Example: List the sample space for each activity or experiment. c) The Saddle Shop carries barrel saddles, roping saddles, and English saddles. Each of these saddle types come with seat sizes of Youth (12-13 ), small adult (14 ), average adult (15 ), and large adult (16 ). Use a tree-diagram to represent the sample space of saddles carried at the store.

5 Probability of an Event A Notation: P(A) Definition: 1. The proportion of times the outcome would occur in a very long series of independent trials. 2. The long run proportion of times an event would occur compared to the total number of observations. To calculate P(A): List all possible outcomes for the event A. List all the possible events in the Sample Space S. Divide: P(A)=

6 a) Use set notation to write the sample space. Example: Consider rolling a pair of die.

7 Example: Consider rolling a pair of die. b) What is the probability that the two dice sum to five?. c) What is the probability that the two dice sum to fourteen?. Null Set ( ) : An impossible event occurring with probability zero.

8 Probability Properties: 1. Probabilities are always a number between 0 and The sum of all outcome probabilities must add to P(A) = 0 if the event never occurs. 4. P(A) = 1 if the event always occurs. 5. Generally when P(A) <.05, the event A can be considered unlikely.

9 Example An automatic transmission can make noises such as whining, whirring, or grinding. The most common possible causes are a clogged filter, a defective pump, a defective torque converter, or defective gears. [Duffy] We will define the four events A, B, C, and D as: a) What is the probability that the noise in the transmission is due to defective gears? Event Action Probability A clogged filter.4 B defective pump.3 C defective torque converter.1 D defective gears? b) Is this a legitimate probability distribution?

10 Example The stem and leaf plot below shows the number of DVDs owned by a sample of 14 homes. If a home is selected at random, find the probability that the home owns more than 54 DVDs. Would this be considered an unusual event?. 1 0, 1 2 0, 0, 7 3 3, 4, 5 4 8, 9 5 3, 5, 7, 9

11 Complement Rule: Use this rule when you know the probability that event A does occur and you want to find the probability that event A does not occur. Notation: Example: What is the compliment of S?

12 Example Dally roping is a style of roping in which the roper throws a half hitch of rope around the saddle horn after a catch is made. The loose end of the rope is held in the roper s hands so he/she can shorten it or let it slip in case of an emergency.... Sometimes when dallying a cowboy/girl will lose a finger because it gets caught between the rope and saddle horn. [Fife] The following table summarizes observations made on 256 ropers. Condition of Fingers U: Undamaged Fingers D: Damaged Finger(s) Position H: Header L: Healer Total M: Missing One or More Fingers Total a) Find the probability that a roper randomly chosen from this sample has undamaged fingers. b) Find the probability that a roper randomly chosen from this sample is not a healer. c) Find the probability that a roper randomly chosen from this sample has undamaged fingers and is a header.

13 Works Cited Duffy, James E. (2000). Modern Automotive Technology. Tinley Park, Illinois: The Goodheart-Willcox Company, Inc.. p , Fife Folklore Archives Home. (2013). AN EXPERIMENT ROPING TERMINOLOGY. Retrieved January 6, 2013, from