Ch. 6: Discrete Probability--Questions
Probability Assignment
• Assignment by intuition – based on intuition, experience, or judgment.
• Assignment by relative frequency – P(A) = Relative Frequency =
• Assignment for equally likely outcomes
nf
Number of Outcomes Favorable to Event ( )Total Number of Outcomes
AP A
One Die• Experimental Probability (Relative Frequency)
– If the class rolled one die 300 times and it came up a “4” 50 times, we’d say P(4)= _____
– The Law of Large numbers would say that our experimental results would approximate our theoretical answer.
• Theoretical Probability– Sample Space (outcomes): 1, 2, 3, 4, 5, 6– P(4) = ____– P(even) = ___
Two Dice
• Experimental Probability– “Team A” problem on the experiment: If we rolled
a sum of “6, 7, 8, or 9” 122 times out of 218 attempts, P(6,7,8, or 9)= 122/218= 56%
– Questions: What sums are possible?– Were all sums equally likely?– Which sums were most likely and why?– Use this to develop a theoretical probability– List some ways you could get a sum of 6…
Outcomes
• For example, to get a sum of 6, you could get:
Two Dice – Theoretical Probability
• Each die has 6 sides.• How many outcomes are there for 2 sides?
(Example: “1, 1”)• Should we count “4,2” and “2,4” separately?
Sample Space for 2 Dice
List the outcomes in the sample space
If Team A= 6, 7, 8, 9, find P(Team A)
Two Dice- Team A/B
• P(Team A)= ___• P(Team B) = ___• Notice that P(Team A)+P(Team B) = ___
Some Probability Rules and Facts
• 0<= P(A) <= 1• Think of some examples where
– P(A)=0 P(A) = 1• The sum of all possible probabilities for an
experiment is 1. Ex: P(Team A)+P(Team B) =1
One Coin
• Experimental– If you tossed one coin 1000 times, and 505 times
came up heads, you’d say P(H)= ___– The Law of Large Numbers would say that this
fraction would approach the theoretical answer as n got larger.
• Theoretical– Since there are only 2 equally likely outcomes,
P(H)= ___
Two Coins
• Experimental Results– P(0 heads) = – P(1 head, 1 tail)=– P(2 heads)=– Note: These all sum to 1.
• Questions:– Why is “1 head” more likely than “2 heads”?
Two Coins- Theoretical Answer
• Outcomes:
2 Coins- Theoretical answer
P(0 heads) = ___P(1 head, 1 tail)= 2/4 = ___P(2 heads)= ___
Note: sum of these outcomes is ___
Three Coins
• Are “1 head” , “2 heads”, and “3 heads” all equally likely?
• Which are most likely and why?
Three Coins1 2 3
3 coins
• P(0 heads)=• P(1 head)= • P(2 heads)=• P(3 heads)=
• Note: sum is ____
Cards• 4 suits, 13 denominations; 4*13=52 cards• picture = J, Q, K
A 2 3 4 5 6 7 8 9 10 J Q KHeart (red)Diamond (red)Clubs (black)Spades (black)
When picking one card, find…
• P(heart)=• P(king)=• P(picture card)=• P(king or queen)=• P(king or heart)=
P(A or B)
• If A and B are mutually exclusive (can’t happen together, as in the king/queen example), then P(A or B)=P(A) + P(B)
• If A and B are NOT mutually exclusive (can happen together, as in the king/heart example), P(A or B)=P(A) + P(B) –P(A and B)
•
P (A and B)
• For independent events: P(A and B)• P(A and B) = P(A) * P(B)
• In General:• P(A and B) = P(A) * P(B/given A)
2 cards (independent) -questions
• Example: Pick two cards, WITH replacement from a deck of cards,
• P(king and king)=• P(2 hearts) =
P(A and B) Example-- Independent• For independent events: P(A and B)• P(A and B) = P(A) * P(B)• Example: Pick two cards, WITH replacement
from a deck of cards, • P(king and king)= ___• P(2 hearts) = ____
P(A and B) – Dependent (without replacement)
• In General:• P(A and B) = P(A) * P(B/given A)• Example: Pick two cards, WITHOUT
replacement from a deck of cards, • P(king and king)= ____• P(heart and heart)= ____• P(king and queen) = ___
Conditional Probability
Wore seat belt
No seat belt Total
Driver survived
412,368 162,527 574,895
Driver died 510 1601 2111
Total 412,878 164,128 577,006
Find: P(driver died)=P(driver died/given no seat belt)=P(no seat belt)= P(no seat belt/given driver died)=
Wore seat belt
No seat belt
Total
Driver survived
412,368 162,527 574,895
Driver died
510 1601 2111
Total 412,878 164,128 577,006
• P(driver died)= ___• P(driver died/given no seat belt)= ___• P(no seat belt)= ___• P(no seat belt/given driver died)= ___