Probability Part 3 – What is Probability?

Probability Warm-up How many ways can a dinner patron select 3 appetizers and 2 vegetables if there are 6 appetizers and 5 vegetables on the menu?

Probability Warm-up How many ways can a jury of 6 women and 6 men be selected from 10 women and 12 men?

Probability Agenda Warm-up Homework Review Objective: To understand and apply the concept of empirical and classical probability. Summary Homework

Probability Models The sample space S of a random phenomenon is the set of all possible outcomes. An event is an outcome or a set of outcomes (subset of the sample space). A probability model is a mathematical description of long-run regularity consisting of a sample space S and a way of assigning probabilities to events.

Probability Model for Two Dice Random phenomenon: roll pair of fair dice. Sample space: Probabilities: each individual outcome has probability 1/36 (.0278) of occurring.

Randomness and Probability Random: individual outcomes are uncertain but there is a regular distribution of outcomes in a large number of repetitions Relative frequency (proportion of occurrences) of an outcome settles down to one value over the long run. That one value is then defined to be the probability of that outcome.

Probability Two Methods of Probability Relative Frequency or Empirical Method P(E)= no. of desired outcomes/no. of trials Classical Method P(E) = No. of ways desired outcome can occur/no. of possible outcomes

Probability Rule 1 Any probability is a number between 0 and 1. A probability can be interpreted as the proportion of times that a certain event can be expected to occur. If the probability of an event is more than 1, then it will occur more than 100% of the time (Impossible!).

Relative-Frequency (Empirical) Probabilities Can be determined (or checked) by observing a long series of independent trials (empirical data) experience with many samples simulation (computers, random number tables)

Probability Example- Each student flip a coin ten time. How many heads? Each student role a die 10 times. How many times did 6 appear?

Relative-Frequency Probabilities Coin flipping:

Probability Rule 2 All possible outcomes together must have probability 1. Because some outcome must occur on every trial, the sum of the probabilities for all possible outcomes must be exactly one. If the sum of all of the probabilities is less than one or greater than one, then the resulting probability model will be incoherent.

Probability Model for Two Dice Random phenomenon: roll pair of fair dice. Sample space: Probabilities: each individual outcome has probability 1/36 (.0278) of occurring.

If two events have no outcomes in common, they are said to be disjoint. The probability that one or the other of two disjoint events occurs is the sum of their individual probabilities. Age of student first entering college under 20: 65% 20-24: 23% 25+: ? } 24 or younger: 88% Rule 3 (or 2): 12% Probability Rule 3

Probability Rule 4 The probability that an event does not occur is 1 minus the probability that the event does occur. (Complementary Event) As a jury member, you assess the probability that the defendant is guilty to be Thus you must also believe the probability the defendant is not guilty is 0.20 in order to be coherent (consistent with yourself). If the probability that a flight will be on time is.70, then the probability it will be late is.30.

Probability Rules: Mathematical Notation

Random phenomenon: roll pair of fair dice and count the number of pips on the up-faces. Find the probability of rolling a 5. P(roll a 5)= P( )+P( )+P( )+P( ) = 1/36 + 1/36 + 1/36 + 1/36 = 4/36 = 0.111

Personal or Subjective Probabilities The degree to which a given individual believes the event in question will happen Personal belief or judgment Used to assign probabilities when it is not feasible to observe outcomes from a long series of trials assigned probabilities must follow established rules of probabilities (between 0 and 1, etc.)

Personal or Subjective Probabilities Examples: probability that an experimental (never performed) surgery will be successful probability that the defendant is guilty in a court case probability that you will receive a ‘B’ in this course probability that your favorite baseball team will win the World Series in 2020

Probability 1) Find the sample space for the gender of the children if a family has three children. Use B for boy and G for girl. BBB BBG BGB GBB GGG GGB GBG BGG 2) A card is drawn from an ordinary deck. Find these probabilities. a. Of getting a jack 4/52 or 1/13 b. Of getting the 6 of clubs (i.e., a 6 and a club) 1/52 c. Of getting a 3 or a diamond 16/52 or 4/13 d. Of getting a 3 or a 6 8/52 3) Find the complement of each event. a. Rolling a die and getting a 4 5/6 b. Selecting a letter of the alphabet and getting a vowel 21/26 c. Selecting a month and getting a month that begins with a J 3/4 d. Selecting a day of the week and getting a weekday 2/7

Probability Summary Empirical & Classical Probability Probability Rules Subjective/Personal Probability

Probability Homework Practice Sheet 3