Wetzel - Central Bucks School District



NAME:____________________ SADA: Probability Notetaker!

Random (trials)-

Probability-

Experimental Probability-

Ex: If I toss a coin 30 times, and get 12 heads, what’s the experimental prob. of getting heads?

Theoretical Probability-

Ex: Using the same coin tossing situation above, what’s the theoretical prob. of getting heads?

Probability Models-

Sample Space-

Example: What’s the sample space for the Tony Gwyn experiment?

How about for the spinner experiment?

What about the rolling 2 dice experiment?

Probability Notation:

• A, B, C, etc. =

• P(A) =

• S =

• When we represent events, we draw them with ____________________

• Venn Diagrams use ____________________________________________

o Examples:

General Set Theory

Union:

• Meaning:

• Symbol:

• Example 1:

• Example 2: Set A = {2, 4, 6, 8, 10, 12}

Set B = {1, 2, 3, 4, 5, 6, 7}

A U B = A or B = { }

Intersection:

• Meaning:

• Symbol:

• Example 1:

• Example 2: Set A = {2, 4, 6, 8, 10, 12} Set B = {1, 2, 3, 4, 5, 6, 7} A [pic] B = A and B =?

Complement:

• Meaning:

• Symbol:

• Example 1: Shade Ac Shade Ac [pic] B

• Example 2: Set A = {2, 4, 6, 8, 10, 12}

S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} =sample space

Ac = { }

TRY THESE:

Sample Space = {1, 2, 3, 4, … 22, 23, 24, 25}

A = {1, 3, 6, 7, 8, 10, 11, 13, 14, 15}

B = {3, 5, 7, 9, 11, 13, 15, 17, 19, 21}

C = {2, 4, 6, 8, 10, 12, 14, 16, 20, 22, 24}

1) What is A [pic] B? 9) What is P(A [pic] B)?

2) What is A U B? 10) What is P(A U B)?

3) What is Bc? 11) What is P(Bc)?

4) What is C [pic] Bc? 12) What is P(C [pic] Bc)?

5) What is A [pic] B [pic] C? 13) What is P(A [pic] B [pic] C)?

6) What is A U B U C? 14) What is P(A U B U C)?

7) What is A [pic] C? 15) What is P(A)?

8) What is Cc? 16) What is P(B)?

Try these…. SET THEORY

S = sample space = {1, 3, 4, 5, 6, 7, 8, 11, 13, 15, 16, 17, 19, 21, 22, 23, 26, 28, 29, 30}

E = {1, 3, 5, 7, 15, 17, 23}

H = {4, 6, 8, 16, 22, 26, 28, 30}

D = {3, 4, 5, 6, 7, 8, 11, 13, 15}

C = {15, 16, 17, 19, 21, 23, 26}

A = {1, 5, 15, 16, 22, 26, 28, 30}

B = {7, 13, 15, 16, 17, 22, 28, 30}

Assuming all data in S are equally likely, find each of the following:

a. P(H) = b. P(E) =

c. E ( H = { d. P(E ( H) =

e. D ( B = { f. P(D ( B) =

g. P(D) = h. P(A) =

i. P(Ac) = j. P(Cc) =

k. P(B) = l. P(E ( B) =

m. P(Dc) = n. P(A ( B) =

o. P(E ( C) = p. P(A ( B) =

q. P(E ( H) = r. P(C ( A) =

Probability Rules

• Let A and B be events

• Let S = sample space

• Let Ac = the complement of event A

List the first 3 probability rules:

(1)

(2)

(3)

Example 1: If the probability of hitting a homerun is 30%, what’s the probability of not hitting a homerun?

P(H) =

P(Hc) =

Example 2: If there are only 8 different blood types, fill in the chart below:

|Type |A+ |A- |B+ |B- |AB+ |AB- |O+ |O- |

|Probability |0.16 |0.14 |0.19 |0.17 |  ? |0.07 |0.1 |0.11 |

Example 3: Las Vegas Zeke, when asked to predict the ACC basketball Champion, follows the modern practice of giving probabilistic predictions. He says, “UNC’s probability of winning is twice Duke’s. NC State and UVA each have probability 0.1 of winning, but Duke’s probability is three times that. Nobody else has a chance.” Has Zeke given a legitimate assignment of probabilities to all the teams in the conference? Why or why not?

Going back to the examples from before…

Ex #1: There are only 8 different blood types, given in the chart below:

|Type |A+ |A- |B+ |B- |AB+ |AB- |O+ |O- |

|Probability |0.16 |0.14 |0.19 |0.17 |0.06 |0.07 |0.1 |0.11 |

What is the probability of being either Type A+ or B-?

What is the probability of being either Type O- or O+?

What is the probability of being either Type AB+ or A+?

So… to find the probability of event A OR event B we….

P(A or B) = P(A U B) =

Ex #2:

We are picking one card out of a standard 52-card deck (no jokers).

What is the probability of picking a diamond?

What is the probability of picking a 3?

What is the probability of picking a diamond or a 3?

What is the probability of picking a black card?

What is the probability of picking a Jack?

What is the probability of picking a black card or a Jack?

So… to find the probability of event A OR event B we….

P(A or B) = P(A U B) =

But why didn’t we do this in EX #1? What was different about EX #1?

This is called… and it means that…

Other examples of events that are disjoint:

Venn Diagrams:

DISJOINT NOT DISJOINT:

RULE for Unions (OR):

EX #3: The probability of hitting a homerun is 30%. What is the probability of hitting 2 HRs in a row?

How about 3 HRs in a row?

How about 4 HRs in a row?

How about 2 HRs then a non-homerun?

EX #4: Picking poker chips from a bag with replacement:

I have a bag that has 10 poker chips in it. 3 are red, 2 are green, and 5 are blue.

What is the probability of picking a red chip and a blue chip?

What is the probability of picking a green chip and a red chip?

What’s the probability of picking a blue chip and a blue chip?

So… to find the probability of event A AND event B we….

P(A and B) = P(A [pic] B) =

EX #5: Picking poker chips from a bag without replacement

I have a bag that has 10 poker chips in it. 3 are red, 2 are green, and 5 are blue.

What is the probability of picking a red and then a blue?

What is the probability of picking a blue and then a blue?

What is the probability of picking a green and then a red?

Conditional Probabilities:

So… to find the probability of event A AND event B we….

P(A and B) = P(A [pic] B) =

But why didn’t we do this in EX #3 and 4? What was different about EX #5?

This is called… and it means that…

Other examples of events that are independent:

RULE for Intersections (AND):

Thinking about conditional probabilities…

Does P(B|A) = P(A|B)?

Think about our poker chip example. Picking poker chips from a bag without replacement

I have a bag that has 10 poker chips in it. 3 are red, 2 are green, and 5 are blue.

What is the probability of picking red given that you picked blue?

What is the probability of picking blue given that you picked red?

So…

Probability rules worksheet NAME:_____________________

1. If P(A) = 0.26 and P(B) = 0.41 and P(A∩B) = 0.1, find the following:

a. P(A U B) =

b. P(B|A) =

c. Are A and B disjoint events? Why or why not?

2. If P(A) = 0.42, P(B) = 0.33 and A and B are independent, what’s the probability of A and B?

3. If P(A) = 0.6 and P(B) = 0.34 and P(B|A) = 0.2, find the following:

a. P(A and B) =

b. P(A or B) =

4. Let the sample space, S = {all whole number from 0 through 19}

Let the event A = {2, 4, 6, 8, 10, 12}

Let the event B = {3, 6, 9, 12, 15, 18}

Let the event C = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

Let the event D = {1, 4, 7, 8, 10, 14, 16, 18}

Find the following:

a. A ∩ B =

b. P(A ∩ B) =

c. Dc =

d. P(C ∩B) =

e. P(A U B) =

f. P(C ∩ D) =

g. P (Cc) =

h. C U A =

Probability rules worksheet- 2 NAME:_____________________

1. If P(A) = 0.45 and P(B) = 0.60 and P(A∩B) = 0.22, find the following:

a. P(A U B) =

b. P(B|A) =

c. Are A and B disjoint events? Why or why not?

2. If P(D) = 0.32, P(R) = 0.13 and D and R are disjoint, what is the probability of D or R?

3. If P(A) = 0.51 and P(B) = 0.28 and P(B|A) = 0.18, find the following:

a. P(A and B) =

b. P(A or B) =

4. Let the sample space, S = {all whole number from 10 through 30}

Let the event A = {12, 14, 16, 18, 20, 22}

Let the event B = {10, 15, 20, 25, 30}

Let the event C = {12, 13, 14, 15, 17, 18, 19, 20, 25, 27}

Let the event D = {11, 21, 23, 24, 26, 28}

Find the following:

i. A ∩ B =

j. P(A ∩ B) =

k. Dc =

l. P(C ∩B) =

m. P(A U B) =

n. P(C ∩ D) =

o. P (Cc) =

p. C U A =

Probability Rules

Let A, B = events and S = Sample Space

- 0 ≤ P(A) ≤ 1

- P(S) = 1

- P(Ac) = 1-P(A)

Union

P(A U B) = P(A) + P(B) – P(A ∩ B)

• If A and B are disjoint, then P(A U B) = P(A) + P(B) because P(A ∩ B) = 0

Intersection

P(A ∩ B) = P(A) * P(B|A)

• If A and B are independent, then P(A ∩ B) = P(A) * P(B) because P(B|A) = P(B)

Conditional Probability

P(B|A) = P(A ∩ B) P(A) > 0

P(A)

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