Essential Question: If you flip a coin 50 times and get a tail every time, what do you think you will get on the 51st time? Why?

 Experiment → process to generate one or more observable outcomes  Sample space → set of all possible outcomes  Tossing coin → [H,T]  Rolling a number cube → [1,2,3,4,5,6]  Event → any outcome or set of outcomes in the sample space  Probability → a number from 0 to 1 (or 0% to 100%) indicating how likely an event is to occur  Probability Distribution → table to display probability of each event

 Example 1: Probability Distribution  100 marbles in a bag – 50 red, 30 blue, 10 yellow, 10 green a) What is the sample space of the experiment? b) Write out a reasonable probability distribution for this experiment c) What is the probability that a blue or green marble will be drawn?

a) Sample space is [red, blue, yellow, green] c) P({blue, green}) = P(blue) + P(green) = = 0.4 Color of marbleRedBlueYellowGreen Probability

 Mutually exclusive events → no outcomes in common  P(E or F) = P(E) + P(F)  Complement → All outcomes that are not contained in the event.  If an event has a probability p, the compliment has probability 1-p

 Example 2: Mutually Exclusive Events  Which of the following pairs are mutually exclusive  E={A,C,E}F={C,S}  E={a vowel}F={1 st 5 letters of alphabet}  E={a vowel}F={C}  What is the complement of the event {A, S}  What is the probability of the event “the spinner does not land on A?” OutcomeASCE Probability

 Independent event → if one event has no effect on the probability of the other event  P(E and F) = P(E) ∙ P(F) Mutually exclusiveIndependent Two possible events for a single trialResults of two or more trials “or”“and” P(E or F) = P(E) + P(F)P(E and F) = P(E) ∙ P(F)

 Example 3: Independent Events  The probability of winning a game is 0.1. Suppose the game is played on two different occasions. What is the probability of: a)Winning both times? b)Losing both times? c)Winning once and losing once?

 Random Variable → a function that assigns a number to each outcome in the sample space of an experiment  Example 4: Roll two number cubes a) Write out the sample space for the experiment b) Find the range of the random variable c) List the outcomes to which the value 7 is assigned

 Expected value → the average value of all outcomes  If we rolled two number cubes 10 times, and their sum were: 8, 5, 8, 6, 11, 11, 3, 9, 9, 7  The more experiments we run, the closer we get to the expected value. If we ran more experiments above, the average would approach 7

 Example 5: Expected Value  A probability distribution for the random variable in the experiment in Example 4 is given below. Find the expected value of the random variable.  Solution: Just multiply each value by its probability, and add Sum of faces Probability

 The expected value is not always in the range of the random variable  Example 6: Expected Value of a Lottery Ticket  The probability distribution for a $1 instant-win lottery ticket is given below. Find the expected value and interpret the result Win$0$3$5$10$20$40$100$400$2500 Probability

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