Probability I’ve often wondered what were the chances of Dorothy’s house falling directly on the wicked Witch of the East?

Why Probability? The problem facing statisticians is to infer what characteristics would remain if random sampling variability were eliminated. To answer this, we need to know what kind of sample results can be expected on the basis of chance alone How about an example from ESP?

Please think of one of these six cards and concentrate on it.

Now, write down the name of your card on a piece of paper, but do not show it to anyone just yet. Continue to concentrate, but do not “force” your thoughts… just relax and hold the image of the card in your mind.

I will now remove the card which I believe is yours from the set of six. Please read what card you have written down on the paper… your card should not be among these.

Uncertainty Probability gives us a way of expressing uncertainty that an event will occur Probabilities are typically presented in several ways –percents (e.g., there a 20% chance of rain today) –proportions (e.g., the prob. of Tom getting a hit is.29) –fractions (e.g., the prob. of getting a “head” is 1/2).

Approaches to Probability There are three approaches to probability –classical (or logical) –empirical relative-frequency –subjective Each approach has advantages and disadvantages and supplements the others

Classical Approach The probability of an event is given by the number of outcomes of interest divided by the total number of possible outcomes under consideration What is the probability of rolling a “3” on a single die? Given there is one outcome of interest (i.e., a “3”) and a total of six possible outcomes under consideration (i.e., a “1,” “2,” “3,” “4,” “5,” and “6”) P(6) = 1/6 =.17

Assumptions of Classical Approach There are two assumptions made when calculating probabilities with the classical approach –each outcome is equally likely –probabilities are based on an infinite series of trials Those assumptions, however, are often not met in real life

A Counting Problem One difficulty encountered is knowing how many possible outcomes there are to consider –For example, given four basketball teams at the local youth center, what’s the probability the end-of-season standings will be 1st place - Bill’s Barbers 2nd place - Mike’s Meats 3rd place - George’s Garage 4th place - Lou’s Laundry Permutations and combinations help answer such questions

Permutations When you have a set of values (e.g., teams) and need to know in how many different ways they can be ordered, you are asking about permutations (order is important) n P n = n! The symbol “!” represents factorial: n(n-1)(n-2)…1 In this example, 4 x 3 x 2 x 1 = 24

Permutations Continued When you have a set of values and need to know in how many different ways some of them can be ordered, you are asking about the permutation of n objects taken r at a time For example, how many different 1st and 2nd place finishes could there be? n P r = n! (n-r)! 4 x 3 x 2 x 1 2 x 1 = 12 In this example, 4! (4-2)! = 4 P 2 =

Combinations When you have a set of values and need to know in how many different ways some of them can be grouped together, you are asking about the combination of n objects taken r at a time (order is not important) For example, given eight students, how many ways could they be put together as lab partners? n C r = n! r!(n-r)! 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 2 x 1 (6 x 5 x 4 x 3 x 2 x 1) = 28 4 In this example, 8! 2! (8-2)! = 8 C 2 =

Empirical Relative-Frequency Approach The probability of an event is given by the number of times the outcome of interest occurs divided by the number of sampling experiments conducted What is the probability of rolling a “3” on a single die? P(3) = 5/24 =.21 Outcome Frequency n = 24

Drawbacks to Empirical Approach The true probability is approached as the number of sampling experiments approaches infinity Data is not always available

Subjective Approach The probability of an event is based on one’s expectations that the event will occur Expectations may arise from personal experience, intuition, expertise, etc May be only means of assigning a probability in some situations

Problems with Subjective Approach Different people are likely to assign different probabilities to the same event

E 1 E 3 E 5 E 2 E 4 E 6 Sample Space for Rolling a Die Basic Concepts Simple event (E i ) - a single outcome (e.g., rolling a “3”) Compound event - an event that can be broken down into two or more simple events (e.g., “an even number”) Sample space - set of all sample points –Assign a sample point to each simple event

Euler Diagram Euler diagrams can be used to illustrate the calculation of probabilities E 1 E 3 E 5 E 2 E 4 E 6 Sample Space for Rolling a Die P(even number) = 3/6 =.5 P(number < 4) = 3/6 =.5 P(even number or number <4) = 5/6 =.83

More Concepts Union of events ( ) - the probability of event A or event B or both A and B P(A) Intersection of A and B Addition Rule: P(A B) = P(A) + P(B) - P(A B)

An Example E 1 E 3 E 5 E 2 E 4 E 6 Sample Space for Tossing a Coin What is the probability of rolling an even number (event A) or a number less than five (event B) or both? P(A B) = 3/6+ 4/6- 2/6 = 5/6 event B event A both A and B

When events are mutually exclusive -- one outcome precludes another outcome from occurring -- the individual probabilities are added –For example, the probability of rolling a number less than 3 (event A) or greater than 4 (event B) Addition Rule for Mutually Exclusive events: P(A B) = P(A) + P(B) = 2/6 + 2/6 = 4/6

Even More Concepts Intersection of events or joint occurrence ( ) - the probability of event A and event B –For example, we might want to know the probability of getting three consecutive “heads” being a psychology major and an internship participant having a heart attack and having no family history of heart disease The calculation of probabilities for the joint occurrence of events depends on whether the events are independent or not.

Independent Events Multiplication Rule for Independent Events: P(A B) = P(A) P(B) For example, the probability of rolling three consecutive “4s” = 1/216 When events are independent -- the outcome of one event has no influence on the outcome of another event -- the individual probabilities are multiplied P(A B C) = P(A) P(B) P(C) = 1/6 x 1/6 x 1/6

There are some instances when the outcomes are not independent and the outcome of one event does influence the outcome of another event Multiplication Rule for Non-independent Events: P(A B) = P(A) P(B|A) = P(B) P(A|B) Intersection of A and B A B

Conditional Probabilities Conditional probability (A|B or B|A) -- the probability of one event occurring given that another other event has occurred. –For example, selecting a card from a deck influences the probability of the next card P(A B) P(B) P(A B) P(A) P(A|B) =P(B|A) =

Illustration of Conditional Probability Lowerclassmen (B) n = 50 Females (A) n = 30 P(A B) P(B) P(A|B) = 100 Students