2. Aim: Intro to Probability Theory Course: Math Lit. Aim: What is the probability of understanding probability? Do Now: How many different possibilities, or outcomes are there with the roll of 1 die? How many cards are there is a deck of playing cards? How many aces?

3. Aim: Intro to Probability Theory Course: Math Lit. Experimental or Empirical Probability Probability -The likelihood an event will occur described as a number between 0 and 1. Experimental or Empirical Probability – gathering data through observations to determine the likelihood of an event occurring, expressed mathematically as a ratio. Ex. Eli Manning completes 30 of 40 passes. The probability of a completion is ¾ or 75%

4. Aim: Intro to Probability Theory Course: Math Lit. Experimental or Empirical Probability A survey of the Registrar of Voters office in a certain city showed the following information on the ages and party affiliations of registered voters. If one voter is chose from this survey, what is the probability that the voter is a Republican? AgeRepubDemoIndepOtherTotal 18-2820543298112847 29-3831130110983804 39-49250251150122773 50 + 272283142107804 Total103812674994243228

5. Aim: Intro to Probability Theory Course: Math Lit. Theoretical Probability Theoretical probability Theoretical probability - the number of ways that an event CAN occur divided by the total number of possible outcomes.  experiment – an occurrence where outcome is not certain  outcome – a result of some activity  sample space – all possible outcomes  event – any subset of the sample space  equally likely/mutually exclusive – chance of all outcomes distinct & the same Probability is expressed as a ratio, decimal or a percent. The law of large numbers – Empirical probability of an event approaches the theoretical probability of an event the more times the experiment is repeated.

6. Aim: Intro to Probability Theory Course: Math Lit. Model Problem A die is rolled once. Find the probability of rolling: a)a 3 b)an even number c)a number less that 5 d)a number less than 10 e)a number greater than 6 Sample space

7. Aim: Intro to Probability Theory Course: Math Lit. The Spinner What is the probability of landing on red? The sum of the theoretical probabilities of all possible outcomes in the sample space is 1. equally likely – chance of all outcomes is the same

8. Aim: Intro to Probability Theory Course: Math Lit. What’s the difference between spinners? UniformBiased or weighted Each event has an equal chance of occurring Each event has an unequal chance of occurring P(Red) = 1/8P(Red) = 2/8 P(Blue) = 1/8P(Blue) = 2/8

9. Aim: Intro to Probability Theory Course: Math Lit. determine sample space AB Model Problem You and an opponent are to spin a spinner simultaneously, and the one with the higher number wins. Which spinner would you choose & why? 168 4 5 7 4, 84, 14, 6 5, 85, 15, 6 7, 87, 17, 6 A B 8 6 1 5 4 7

10. Aim: Intro to Probability Theory Course: Math Lit. Model Problem What is the probability of rolling a pair of dice and getting a twelve?

11. Aim: Intro to Probability Theory Course: Math Lit. Probability & Set Theory Sample space: n outcomes Event A: m outcome In a class of 147 students, 95 are taking math (M), 73 are taking science (S) and 52 are taking both. MS435221 31

12. Aim: Intro to Probability Theory Course: Math Lit. Model Problem In a class of 147 students, 95 are taking math (M), 73 are taking science (S) and 52 are taking both. MS435221 What is probability student takes only Math? 31

13. Aim: Intro to Probability Theory Course: Math Lit. Geometric Probability 20 10 5 2 1 What must we know to find the probability of scoring a 20 on the toss of a dart? 5r5r The areas of the circle and each of the rings If the radius of the circle is 5r and each ring is r wide, what is the probability of scoring 20 on one toss?

14. Aim: Intro to Probability Theory Course: Math Lit. Numerical Values of Probability The values of probability ratios range from 0 through 1. (fractions, decimals, %) A probability of 0 means that the outcome will never happen. A probability of 1 means that the outcome will be certain to happen. The closer a probability is to 0, the less likely the outcome. The probability of event (A) plus the probability of "not A” or ~A, equals 1: P(A) + P(~A) = 1; P(A) = 1 – P(~A) Values for a Probability Ratio

15. Aim: Intro to Probability Theory Course: Math Lit. Numerical Values of Probability The probability of an event E must be equal to or greater than zero (0) and less than or equal to one (1). Values for a Probability Ratio 0 1.5 Always Even Chance NeverNot Likely Likely.75.25 What is the probability of rolling 13 with a pair of dice? 0

16. Aim: Intro to Probability Theory Course: Math Lit. Model Problem A card is chosen at random. Find the probability a)the card is an ace b)the card is a heart c)the card is a face card d)the card has a number value less than 6

17. Aim: Intro to Probability Theory Course: Math Lit. Model Problems A bag contains a nickel, a dime and a quarter. A person selects one of the coins. What is the probability that the coin is worth A. Exactly ten cents B. Exactly three cents C. More than three cents. P(coin is worth.10) = 1/3 P(coin is worth.03) = 0/3 P(coin is worth >.03) = 3/3 = 1

18. Aim: Intro to Probability Theory Course: Math Lit. Model Problem A letter is chosen at random from the word REED. A. List the sample space. B. Find the probability of choosing an E. A. {R, E, E, D} P(E) = Number of E’s Number of letters P(E) = 2 4 or 1/2 B.

19. Aim: Intro to Probability Theory Course: Math Lit. Model Problems 1. An urn (jar) contains 4 red marbles and 6 white marbles. If one marble is drawn at random, what is the probability that it will be blue? 2. The probability that we will win the baseball game is x/5. What is the probability that we will not win the game? 3. If t represents the probability the event T will occur, what is the probability that event ~T will not occur?

20. Aim: Intro to Probability Theory Course: Math Lit. Model Problem In the Sullivan family, there are two more girls than boys. At random, Mrs. Sullivan asks one of her children to go to the store. If she is equally likely to ask any one of her children, and the probability that she asks a girl is 2/3, how many boys and how many girls are there is the Sullivan family? x = number of boys x + 2 = number of girls 2x + 2 = number of children P(girl) = Number of girls Number of children x + 2 2x + 2 2 3 = 4x + 4 = 3x + 6 x = 2 2 4

21. Aim: Intro to Probability Theory Course: Math Lit. Model Problem Five more men than women are riding a bus as passengers. The probability that a man will be the first passenger to leave the bus is 2/3. How many passengers on the bus are men, and how many are women? x = number of women x + 5 = number of men 2x + 5 = number of passengers P(man) = Number of men Number of passengers x + 5 2x + 5 2 3 = 4x + 10 = 3x + 15 x = 5 x + 5 = 10 5 10

22. Aim: Intro to Probability Theory Course: Math Lit. The Product Rule