Download Ch 6.3 General Probability Rules

AP STATISTICS
Types of Probability

 Theoretical: true mathematical probability
 Empirical: the relative frequency with which an
event occurs in a given experiment
 Subjective: an educated guess
Review Terms

 Experiment: any process that yields a result or
observation
 Outcome: a particular result of an experiment
 Sample Space: the collection of all possible outcomes
 Event: any collection of outcomes; any subset of the
sample space
 Example: Roll die
 Let A={1}
 Let B={2, 4, 6}
 Let C={3, 6}
 An event occurs if any outcome of the event occurs
Probability of Event

 P(A) = # of outcomes/# outcomes in sample
space
 If each outcome is equally likely
 Using previous slide’s events:
 P(A)=
 P(B)=
 P(C)=
Rules of Probability

 Rule 1: For any event, 0 ≤ 𝑃(𝐴) ≤ 1
 Rule 2: Complement Rule
 “A complement” is the event that A does not occur;
the set of all outcomes not in A
 𝑃 𝐴 + 𝑃 𝐴′ = 1
 Example: even/odd; alive/dead
 P(ACE)=
 P(ACE’)=
Rules of Probability

 Rule 3: Mutually Exclusive/disjoint-two events, A &
B that have no outcomes in common
 Examples: red and spade; freshman, sophomore,
junior
 Note: complementary→mutually exclusive ;
mutually exclusive→complementary
 Rule 4: Additive Rule for disjoint events
 𝑃 𝐴∪𝐵 =𝑃 𝐴 +𝑃 𝐵
1
2
1
4
 Example: P(red or spade)=P(red) + P(spade)= + =
3
4
Rules of Probability

 Rule 5: Additive Rule-For any events A & B,
P(A or B)=P(A)+P(B)-P(A and B)
Example: Find the probability of drawing red card or
Ace. 𝑃 𝑟𝑒𝑑 ∪ 𝐴𝑐𝑒 = 𝑃 𝑟𝑒𝑑 + 𝑃 𝐴𝑐𝑒 − 𝑃 𝑟𝑒𝑑 𝐴𝑐𝑒
=
26
4
2
+ −
52
52
52
=
28
52
 Rule 6: Conditional Probability-For two events A and
B, the probability that A occurs given that B has
occurred.
𝑃 𝐴𝐵 =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
Probability Rules

𝑃 𝐴𝐵 =
𝑃(𝐴∩𝐵)
𝑃(𝐵)
 Example: What is the probability that a card is a diamond,
given that it is red?
𝑃(𝐷 ∩ 𝑅)
13/52 13
𝑃 𝐷𝑅 =
=
=
= .5
𝑃(𝑅)
26/52 26
 Example: What is the probability of 2, given that you got
an even number?
𝑃(2 ∩ 𝐸)
1/6
𝑃 2𝐸 =
=
= 1/3
𝑃(𝐸)
3/6
Independent Events

 2 events A and B are independent if the occurrence of one event
doesn’t affect the probability of occurrence of the other event
 To prove:
 𝑃 𝐴𝐵 =𝑃 𝐴
𝑜𝑟 𝑃 𝐵 𝐴 = 𝑃 𝐵
 Example: rolling two dice; drawing from a deck with replacement
 Are drawing a face card and drawing a red card independent
events?
𝑃 𝐹𝑅 =𝑃 𝐹
𝑃(𝐹 ∩ 𝑅)
= 12/52
𝑃(𝑅)
6/52
= 12/52
26/52
So they are independent.
Probability Rules

 Rule 7: Multiplicative Rule for Independent Events𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃(𝐵)
 Has to be shown or given
 Example: Find the probability of drawing two Queens from
a deck of cards if it is done with replacement.
4 4
𝑃 𝑄1 ∩ 𝑄2 = 𝑃 𝑄1 𝑃 𝑄2 =
∗
≈ .0059
52 52
 Rule 8: Multiplicative Rule- For any two events A& B, the
𝑃 𝐴 ∩ 𝐵 = 𝑃 𝐴 𝑃 𝐵 𝐴 or 𝑃 𝐵 = 𝑃(𝐵)𝑃 𝐴 𝐵
 Example: Find the probability of drawing two Queens from
a deck of cards if it is done without replacement
4 3
𝑃 𝑄1 ∩ 𝑄2 = 𝑃 𝑄1 𝑃 𝑄2 𝑄1 =
∗
≈ .0045
52 51
Example: Contingency Table
G
PG
PG-13
R
2000s
2
13
22
2
1990s
1
1
7
0
1980s
0
1
0
0
1970s
0
1
0
0
9
 P(1990’s)=
50
29
 P(PG-13)=
50

= .18
= .58
7
50
 P(1990s and PG-13)=
= .14
 Are 1990s and PG-13 disjoint events?
G
PG
PG-13
R
2000s
2
13
22
2
1990s
1
1
7
0
1980s
0
1
0
0
1970s
0
1
0
0

 The probability that a randomly selected DVD is rated
PG-13 or is from the 1990s.
 P(PG-13 or 1990s)=P(PG13)+P(90s)-P(PG13 and 90s)
=.58+.18-.14=.62
 𝑃 90𝑠 𝑃𝐺13 =
𝑃(90𝑠 ∩𝑃𝐺13)
𝑃(𝑃𝐺13)
=
7/50
29/50
= 7/29
 Are PG-13 and 1990s independent or dependent events?
𝑃 90𝑠 𝑃𝐺13 ≠ 𝑃(90𝑠) dependent
Other ways:
𝑃 𝑃𝐺13 = 𝑃 𝑃𝐺13 90𝑠
𝑃 𝑃𝐺13 𝑎𝑛𝑑 90𝑠 = 𝑃 𝑃𝐺13 𝑃(90𝑠)
Homework

 6.69, 71, 78
Have a wonderful weekend!