James Logan

AP Statistics High School Mr. Baumback

4.2: Probability Models

Four Coins Game Work in pairs:

Flipper & Recorder

Flipper: simulates 4 coin toss on calculator

Recorder: records 16 results

0 = Head = $1

Construct table:

1 = Tail = -$1

X -4 -2 0 2 4

frequency

P(x)

Results Empirical Distribution X -4 -2 0 2 4 total Terms: event: outcome: sample space:

Count

P (X)

Probability Model

Ex.: roll a die

• List all outcomes for x

X P(x)

• 0 ≤ P(x) ≤ 1

1 1/6

• ∑P(x) = 1

2 1/6 3 1/6 4 1/6 5 1/6 6 1/6 1

Probability Rules P(

)=?

OR

AND

Add

Multiply

Disjoint ?

Independent ?

Addition

Multiplication

P(A or B) = P(A) + P(B) if disjoint P(A or B) = P(A) + P(B) – P(A & B) if not disjoint

P(A & B) = P(A) ⋅ P(B) if A & B are independent (the chance of one doesn’t affect the chance of the other)

€ A

B disjoint

not disjoint

Complement P(A) = 1 – P(Ac) where Ac is the complement of A

A

Ac

Ex: Mutually Exclusive or Not Mutually Exclusive? event A

event B

sum = 8 sum = 12 sum = 7 sum = 7 doubles sum = 8

sum = 10 doubles sum is even sum is odd sum is odd sum is even

mutualy exclusive (disjoint)

Ex: Indepndent or Not Independent? events

independent or not independent

Toss a die twice. The 1st is a 6, the 2nd is a 3. Draw an ace from a deck, then draw a 2nd card. Draw a Jack of Hearts, replace it and draw another card.

Ex:

From a regular deck of playing cards two cards are drawn at random without replacement. A. P( an ace and then a king ) =

B. P( exactly one face card ) =

C. P( at least one face card ) =

Ex: One Die is Rolled Twice. Find . . .

A. P( both rolls are odd ) =

B. P( both rolls are 3’s ) =

C. P( one roll is a 5 , the other is a 4 ) =

D. P( the sum is 7 or both rolls will show the same number ) =

Ex:

Toss a coin four times. A. P( 4 heads ) =

B. P( at least 3 heads ) =

C P( at least 1 head ) =

Ex: Pick a Key The probability that a certain door is locked is .5. The key to the door is one of ten unidentified keys hanging from a key rack. Two keys are selected at random before approaching the door. Find the probability that the door opens before returning for a 3rd key.

Ex: Mathlete Contest Problem- Bishop O’Dowd HS, Novemeber 18, 1992

Problem #3 (3 minutes): Four boxes each contain 5 objects. The first contains 1 red object and 4 white objects, the second 2 red and 3 white, the third 3 red and 2 white, and the fourth 4 red and 1 white. One of the boxes is chosen at random and two objects are drawn from it. What is the probability that both objects are white ?