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 greena)What is the sample space of the

experiment?b)Write out a reasonable probability

distribution for this experimentc)What is the probability that a blue or

green marble will be drawn?

a) Sample space is [red, blue, yellow, green]b)

c) P({blue, green}) = P(blue) + P(green)= 0.3 + 0.1 = 0.4

Color of marble

Red Blue Yellow Green

Probability50

0.5100

300.3

100

100.1

100

100.1

100

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 pairsare mutually exclusive E={A,C,E}F={C,S} E={a vowel} F={1st 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?”

Outcome A S C E

Probability 0.4

0.3

0.2

0.1

Independent event → if one event has no effect on the probability of the other event P(E and F) = P(E) ∙ P(F)Mutually exclusive Independent

Two possible events for a single trial

Results 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?

0.1 0.1 0.01

0.9 0.9 0.81

0.1 0.9 0.9 0.1 0.18

1 (0.01 0.81) 0.18or

Random Variable → a function that assigns a number to each outcome in the sample space of an experiment

Example 4: Roll two number cubesa) Write out the sample space for the

experimentb) Find the range of the random variablec) 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

8 5 8 6 11 11 3 9 9 77.7

10

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

2 3 4 5 6 7 8 9 10 11 12

Probability

136

118

112

19

536

16

536

19

112

118

136

1 1 1 1 5 1 5 1 1 1 12 3 4 5 6 7 8 9 10 11 12 736 18 12 9 36 6 36 9 12 18 36

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

0.882746

0.06

0.04

0.01

0.005

0.002

0.0002

0.00005

0.0000040(0.882746) 3(0.06) 5(0.04) 10(0.01) 20(0.005) 40(0.002)

100(0.0002) 400(0.00005) 2500(0.000004) 0.71

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