1-1

Objectives

Introduce students to the basic concepts of probability, including: randomness combined probability dependent events independent events gambler’s fallacy

1-2

Defining Probability

Probability is A. the likelihood that a given

event will occurB. A number expressing the

likelihood that a specific event will occur, expressed

as the ratio of the number of actual occurrences to the number of possible occurrences (Soukhanov, 1992)

1-3

Probability of rolling a “1”

(# of favorable results) (# of possible results) = 1/6

A "favorable result" is the result for which you are determining the probability. In this case, you are determining the probability of a single event1/6 = .17 = 17%

1-4

Randomness

Each possible outcome in that process has the same chance, or probability, of occurring

Outcomes are determined by chance

1-5

Probability in the Real World

Probability = # Favorable Outcomes

Total # Trials

Probabilities based on trials are an estimate

1-6

Combined Probability

Probability of the 1st event AND 2nd event occurring

Probability of 1st event X

Probability of 2nd event

Combined Probability

1-7

Is an Event Dependent or Independent?

INDEPENDENTThe outcome of the 1st event does not affect the outcome

of the 2nd event

DEPENDENTThe outcome of the 1st event

does affect the outcome of the 2nd event

1-8

Gambler’s Fallacy

Mistaking independent events for dependent

events

In reality, the outcomes are completely independent

1-9

Probability of One Event vs. a Group of Events

Assumption: Outcome is Random

Probability of 1st x 2nd x 3rd x 4th =Combined Probability of a group of events

1-10

Shared Birthdays

With any group of 23 people, there is a 50% chance that two or more people in that group will have the same birthday

1-11

Heads or Tails?

In your group, flip the coin 20 times

Each group will record the results

count the number of heads and tails

calculate the percentage of heads

1-12

Outcome and Probability

What are all the possible outcomes for flipping two coins? heads/heads heads/tails tails/heads tails/tails

What are the probabilities of each outcome?

1-13

Becoming a Legend

Batting Average calculated

# Hits during seasonTotal # times at bat

.400 batting avg. meansa hit 4 out of 10 times at bat

1941: Ted Williams approached the last day of play with a .39955 (179/448)The Red Sox coach offered to let Williams sit out the last day to avoid slipping below .400

1-14

Becoming a Legend: Part II

Given Ted Williams batting average before the last day (.39955), how many hits would you expect him to get out of 8 times at bat?

If Williams went to bat 8 times on his final day and got the number of hits calculated above, what would his season average have been?

Would you have played the last games or sat them out?