Probability and Sample Space…….

Vocabulary To Know Probability Experiment: A chance process that leads to well defined results called outcomes. Outcome: The result of a single trial of a probability experiment.

Sample Space Sample Space: the set of ALL possible outcomes of a probability experiment. Example: Flipping a coin has 2 possible outcomes 1. Heads 2. Tails

You Try Find the sample space for the following probability experiments. Toss One Coin Roll a Die Answer a T/F Question Toss 2 Coins

Answers Toss One Coin Roll a Die Answer T/F Question Toss 2 Coins H,T 1,2,3,4,5,6 T,F HH, TT, HT, TH

Finding a Probability The probability of an event can be obtained by 3 different methods: Empirical - experimental Theoretical – assumes all outcomes in the sample space are equally likely to occur. Subjective – a value based on an educated guess.

Empirical Probability

Empirical Probability Example If a person rolls a die 40 times and 9 of the rolls results in a “5”, what empirical probability was observed for the event “5”? Answer:

Theoretical Probability

Theoretical Probability Example What is the probability of rolling a die and getting a “5”? Answer:

The difference between theoretical and empirical probability is that theoretical assumes that certain outcomes are equally likely while empirical probability relies on actual experience to determine the likelihood of outcomes.

Law of Large Numbers The Law of Large Numbers says that as the # of trials in an experiment increases, the empirical probability approaches the theoretical probability. If an experiment is done many times, everything tends to “even out.”

Labs Let’s try to see how the Law of Large Numbers works……..

Theoretical Probability…..

How are probabilities expressed? Probabilities are expressed as reduced fractions, decimals rounded to 2 or 3 decimal places, or, where appropriate, percentages Examples: 1. 2. 0.5 3. 50%

Example…… Find the probability of drawing a queen from a deck of cards. Answer:

Example…… If a family has 3 children, find the probability that all 3 children are girls. You are going to have to look at the sample space before you can answer this one.

Looking for all 3 girls…… Sample Space: BBB BBG BGB GBB GGG GGB GBG BGG Answer:

Example…… A card is drawn from an ordinary deck. Find these probabilities: a. P(Jack) b. P(6 of Clubs) c. P(Red Queen)

Answers…… a. b. c.

Probability Rules…… Rule 1: The probability of an event is between 0 and 1. In other words…. *The probability can NOT be negative. *The probability can NOT be greater than 1.

Rule 2: If an event can NOT occur, then the probability is 0. Example: Find the P(9) on a die. Answer: P(9) = 0

Rule 3: If an event is certain, then the probability is 1. Example: Roll a die. What is the probability of getting a number less than 7? Answer: P(# less than 7) = 1

The sum of the probabilities in the sample space is 1. Example: Rule 4: The sum of the probabilities in the sample space is 1. Example: In a roll of a die, each outcome in the sample space has a probability of 1/6. See chart. x 1 2 3 4 5 6 P(x) 1/6 6/6 = 1

Complement…… The complement is the set of all outcomes in the sample space that are NOT included in the event, A. In other words, it is the probability of event NOT occurring.

Example…… Find the complement of getting an odd # on the roll of a die. Answer: Getting an EVEN number.

Example…… If the probability that a person owns a computer is 0.70, find the probability that a person does not own a computer. Answer: P(Not Owning) = 1 -.70 P(Not Owning) = .30

Example…… If the probability that a person does not own a TV is 1/5, find the probability that a person does own a TV. Answer: P(Does) = 1 – 1/5 P(Does) = 4/5

Example…… 2 dice are rolled. Find a. P(sum of 3) b. P(at least 3) c. P(more than 9)

You need your array of the sums first…… 1 2 3 4 5 6 7 8 9 10 11 12

P(sum of 3) 1 2 3 4 5 6 7 8 9 10 11 12 Answer:

P(at least 3) - Use the complement…… 1 2 3 4 5 6 7 8 9 10 11 12 Answer: Prob = 1 – 1/36 = 35/36

P(more than 9)…… 1 2 3 4 5 6 7 8 9 10 11 12 Answer: