Probability Rules Monty Hall Lets Make a Deal

Probability Rules

Monty Hall “Let’s Make a Deal” • Suppose you’re on a game show, and you’re given the choice of three doors. Behind one door is a car, behind the others, goats. • You pick a door, say #1, and the host, who knows what’s behind the doors, opens another door, say #3, which has a goat. • You have the option to stay with your original door choice or to switch to #2. Is it to your advantage to switch?

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“Let’s Make a Deal” 1. Make a prediction with your partner about what you think we should do. 2. Three cards (two of one color= goats, other color= car) 3. Designate one person to be the game show host (you’ll switch roles eventually) 4. The host should shuffle and arrange the cards so that only they know where the ‘car’ is. 5. Contestant then picks a door. 6. Host reveals one of the doors that wasn’t picked and that has a goat. 7. Contestant decide one of three things: 1) switch 2) stay 3) flip a coin. H=switch, T=stay 8. Record data

Data Collection Switch 5 times, Stick 5 times, flip my coin 5 times Trial 1 2 … 15 trials total person Stick/Switch/Coin Flip & Sw or St Win/Lose

• The host must always open a door that was not picked by the contestant • The host must always open a door to reveal a goat and never the car. • The host must always offer the chance to switch between the originally chosen door and the remaining closed door.

• An intuitive explanation is that, if the contestant initially picks a goat (2 of 3 doors), the contestant will win the car by switching because the other goat can no longer be picked, whereas if the contestant initially picks the car (1 of 3 doors), the contestant will not win the car by switching

• Most people come to the conclusion that switching does not matter because there are two unopened doors and one car and that it is a 50/50 choice. This would be true if the host opens a door randomly, but that is not the case; the door opened depends on the player's initial choice, so the assumption of independence does not hold

Vocab • Probability Model • Sample Space • Event

How could we find S? 1. 2. 3. 4. Make an exhaustive list. Use counting principles. Run a simulation. Somebody gives it to us!

Let’s consider the following example: What is the sample space for the genders of a family consisting of exactly three children?

Toss a Fair Coin • Find the sample space for tossing a fair coin four times. • Find the probability model for # of tails • Find P(0), P(>2)

Probability Rules • The probability of any event is a # btwn 0 and 1 • All possible outcomes together must have probability 1 • The prob. That an event does NOT occur is 1 - prob that is does occur (complement) • If two events have no outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities. • If all outcomes in the SS are equally likely, the prob that event A occurs can be found using formula: • P(A)= # of outcomes corresp to event A/ Total number of outcomes

Tossing 2 Fair 6 -sided Dice • Imagine rolling these dice- one that’s red and one that’s green. • What is the sample space? • What is probability of each outcome? • P(rolling a 5)? • P(rolling not a 5)?

Marital Status • Woman aged 25 -29 years old and their marital status Marital Status Never Married Widowed Divorced Probability 0. 506 0. 452 0. 002 0. 04\ • P(Widowed)? • P(Not married)?

Two-Way Tables • Standard deck of cards (no jokers)= 52 cards • A= getting a face card and B= getting a heart Heart Face card Not a face card Not a Heart

Heart Face card Not a face card 1. P(A)=? 2. P(B)=? 3. P(A and B)=? 4. P(A or B)=? Not a Heart

Rules Again • P(A or B) = P(A)+P(B)- P(A and B) • P(A and B)= P(A)*P(BA)

Two-Way Tables • • Pierced ears? Yes PE No PE Total Male 19 71 90 Female 84 4 88 Total 103 75 178 A= pierced ears 1. P(A)? 2. P(A and B)? 3. P(B and A)? 4. P(A or B)? B= is a male

Two-Way Tables • Shuffle a standard deck of playing cards and deal one card. Let event J= getting a jack and event R=getting a red card. 1. Construct a two-way table that describes the sample space in terms of events J and R 2. Find P(J), P(R), P(J or R), and P(J and R)

Venn Diagram • Standard deck of cards (no jokers)= 52 cards • A= getting a face card and B= getting a heart

Venn Diagram Yes PE No PE Total Male 19 71 90 Female 84 4 88 Total 103 75 178

Vocab • Mutually Exclusive/Disjoint Getting a Heart Getting a Black Card

Genetics at Work • Try to curl their tongue • Interlock the fingers of your hands. Note which thumb (right or left) comes out on top. • Gather data from 12 of your classmates (yourself included). Make sure you keep their data together and label with their initials (e. g. , EM yes curl, left on top) • T= can curl tongue and R= right thumb on top

Genetics 1. Make a Venn diagram that shows where each member of your set of data falls with respect to the two events. Use initials to represent each student. 2. Make a two-way table that summarizes the number of students in each of the four regions of the Venn diagram 3. Find P(T) 4. Find P(T and R) 5. Find P(T or R) 6. Find P(Tc) 7. Find P(Tc and Rc) 8. Challenge: Consider the event F= is female. See if you can construct a Venn diagram with three circles to classify the members of your class with respect to events T, R, and F.