1. Good afternoon!
August 9, 2017

2. Bellringer
x 5.2 =
x 3.5 =

3. Concept Check
___Probability___ ___Venn Diagram___ ___Complement___ ___Independent___ ___Addition Rule___

4. Standard: MGSE9-12.S.CP.2 Understand that if two events A and B are independent, the probability of A and B occurring together is the product of their probabilities, and that if the probability of two events A and B occurring together is the product of their probabilities, the two events are independent. Learning Target: I CAN represent real world objects algebraically. I CAN communicate mathematically using set notation. I CAN make everyday decisions by understanding conditioned probability.

5. Experimenting With Probabilities

6. Vocabulary Independent – event does not affect the 2nd event
Dependent – event does affect the 2nd event
Mutually Exclusive – two events are mutually exclusive if they both cannot occur at the same time

7. Activity – Experiment 1 You roll two fair six-sided dice.
What is the probability that a 4 is rolled with the first die?
What is the probability that a 4 was rolled on the second die?
What is the probability that a 4 show on each die?
P(4) = 1/6 = 0.166
P(4) = 1/6 = 0.166
P1(4)  P2(4) = 1/6  1/6 = 1/36 = .0278

8. Activity – Experiment 2
You pick two cards from a standard deck of 52 cards without replacing the first card before picking the second. What is the probability of a red suited card on the first draw?
What is the probability of a red suited card on the second draw, given that we got one on the first draw?
 What is the probability that the cards are both red suited?
Why are these two experiments different?
P(RS) = 26/52 = 0.5
P2(RS) = 25/51 =
P(both RS) = P1(RS)  P2(RS) = 0.5  =
In the first experiment, events are independent. Not so in the second – events are dependent.

9. Independent vs Dependent
Events A and B are independent if the occurrence of either event does not affect the probability of the occurrence of the other event
Events A and B are dependent if the occurrence of either event does affect the probability of the occurrence of the other event

10. Independent Examples
What is the probability of rolling 2 consecutive sixes on a ten-sided die? What is the probability of flipping a coin and getting 3 tails in consecutive flips?
P(6 and 6) = P(6)  P(6)
= 1/10  1/10
= 1/(102) = 1/100 = 0.01
P(H and H and H) = P(H)  P(H)  P(H)
= 1/2  1/2  1/2
= 1/(23) = 1/8 = 0.125

11. Dependent Examples
What is the probability of drawing two aces from a standard 52-card deck? What is the probability of drawing two hearts from a standard 52-card deck?
P(A1 and A2) = P(A1)  P(A2)
= 4/52  3/51
= 12/2652 = 0.005
P(H1 and H2) = P(H1)  P(H2)
= 12/52  11/51
= 132/2652 = 0.050

12. Venn Diagrams in Probability
A  B is read A union B and is both events combined also seen as A or B
A  B is read A intersection B and is the outcomes they have in common also seen as A and B
Disjoint events have no outcomes in common and are also called mutually exclusive
In set notation: A  B =  (empty set) A B A B
Mutually Exclusive
Intersection: A  B

13. Multiplication Rule: Independent Events
If A and B are independent events, then P(A and B) = P(A) ∙ P(B)
If events E, F, G, ….. are independent, then P(E and F and G and …..) = P(E) ∙ P(F) ∙ P(G) ∙ ……

14. Addition Rule: Disjoint Events
If E and F are disjoint (mutually exclusive) events, then P(E or F) = P(E) + P(F) E F
Probability for Disjoint Events
P(E or F) = P(E) + P(F)

15. Addition Rule for Disjoint Events
If events A, B, and C are disjoint in the sense that no two have any outcomes in common, then
P(A or B or C) = P(A) + P(B) + P(C)
This rule extends to any number of disjoint events

16. Mutually Exclusive Examples
What is the probability of rolling a 6 or an odd-number on a six-sided die? What is the probability of drawing a king, queen or jack (a face card) from a standard 52-card deck?
P(6 or odd) = P(6) + P(odd)
= 1/6 + 3/6
= 4/6 = 2/3 =
P(K or Q or J) = P(K) + P(Q) + P(J)
= 4/52 + 4/52 + 4/52
= 12/52 = 3/13 = 0.231

17. General Addition Rule
For any two events E and F, P(E or F) = P(E) + P(F) – P(E and F) E F
E and F
Probability for non-Disjoint Events
P(E or F) = P(E) + P(F) – P(E and F)

18. General Addition Examples
What is the probability of rolling a 5 or an odd-number on a six-sided die? What is the probability of drawing a red card or a face card from a standard 52-card deck?
P(5 or odd) = P(5) + P(odd) – P(5 and odd)
= 1/6 + 3/6 = 4/6 = 2/3 =
P(R or FC) = P(Red) + P(FC) – P(RFC)
= 12/ /52 – ( 6/52 )
= 32/52 =

19. Summary and Homework Summary
Events are independent if the occurrence of either event does not affect the occurrence of the other
P(A and B) = P(A)  P(B)
Events are dependent if the occurrence of either event does affect the occurrence of the other
P(A and B) = P(A)  P(B|A) (remember for next lesson)
Mutually exclusive events can’t happen at same time
P(A or B) = P(A) + P(B) (P(A and B) = 0)
Events not mutually exclusive
P(A or B) = P(A) + P(B) – P(A and B)