Basic Probabilities Starting Unit 6 Today!

Definitions  Experiment – any process that generates one or more observable outcomes  Sample Space – set of all possible outcomes of the experiment

Examples ExperimentSample Space Tossing a coinHead and tails, written as {H, T} Rolling a dice{1, 2, 3, 4, 5, 6} Choosing a name from a phone book All the names in the phone book Counting the number of fish in Lake Norman The set of non-negative integers

Definitions  Event – any outcome or set of outcomes in the sample space For example, in the experiment of rolling a dice, the set {1, 2, 5} is an event…meaning you rolled a 1, a 2, and a 5.  Probability – how likely the event is to occur A probability of 0 means the even cannot occur A probability of 100 means the even must occur

Probability Distribution  A table describing all probabilities from the experiment.  The sum of all probabilities should equal 1 or 100% of the data.

Probability Distributions  Suppose that 100 marbles are placed in a bag; 50 red, 30 blue, 10 yellow, and 10 green. An experiment consists of drawing one marble out of the bag and observing the color. What is the sample space? Write out a reasonable probability distribution for this experiment and verify that the sum of the probabilities is 1. What is the probability that a blue or a green marble will be drawn?

Mutually Exclusive  Two events are mutually exclusive if they have no outcomes in common. For example, the probability of rolling an even number and the probability of rolling a 1. P(E or F) = P(E) + P(F)  The compliment of an event is the probability that it doesn’t happen 1-probability that it happens

Example a) Which of the following pairs of events E and F are mutually exclusive?  E = {A, C, E}F = {C, S}  E = {a vowel}F = {in the first 4 letters of the alphabet}  E = {a vowel}F = {C} b) What is the compliment of the event {A, S}? c) What is the probability of the event “the spinner does not land on A?” Out- come Prob. A0.4 S0.3 C0.2 E0.1

Independent Events  Two events are independent if the occurrence or non-occurrence of one event has no effect on the probability of another event. P(E and F) = P(E) · P(F)

Mutually Exclusive v. Independent Mutually ExclusiveIndependent Refers to two possible results for a single trial of an experiment. Refers to the results from two or more trials of an experiment. ORAND P(E or F) = P(E) + P(F) P(E and F) = P(E) · P(F)

Example  The probability of winning a certain game is 0.1. Suppose the game is played on two different occasions. What is the probability of: Winning both times? Losing both times? Winning once and losing once?

Random Variables  A random variable is a function that assigns a number to each outcome in the same space of an experiment. For example, heads = 1, tails = 0

Example  An experiment consists of rolling 2 dice. A random variable assigns to each outcome the total of the faces shown. Write out the same space for the experiment. Find the range of the random variable List the outcomes to which the value 7 is assigned.

Answer   The smallest value is 2, which is assigned to (1,1) and the highest is 12 (6, 6). The range is the set of integers from 2 to 12.  The value of 7 is assigned to (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1)

Expected Value  The expected value, or mean, of a random variable is the average value of the outcomes.  As the experiment is tested a large number of times the average will be the expected outcome. For example, flipping a coin…the more times you do it the closer your results are to 50% heads and 50% tails.

Example  A probability distribution for a $1 instant-win lottery ticket is given below. Find the expected value and interpret the results. Win$0$3$5$10$20$40$100$400$2500 Prob

Homework  Worksheet (1-24)