Transcript No Slide Title

ARE YOU READY FOR THE QUIZ?
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Yes
No
That depends on how you
measure ‘readiness’.
33%
33%
33%
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UPCOMING IN CLASS

Monday HW4
CHAPTER 14
From Randomness to Probability
MODELING PROBABILITY (CONT.)

The probability of an event is the number of
outcomes in the event divided by the total
number of possible outcomes.
P(A) =
# of outcomes in A
# of possible outcomes
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FORMAL PROBABILITY
1.
Two requirements for a probability:


2.
A probability is a number between 0 and 1.
For any event A, 0 ≤ P(A) ≤ 1.
Probability Assignment Rule:


The probability of the set of all possible outcomes
of a trial must be 1.
P(S) = 1 (S represents the set of all possible
outcomes.)
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FORMAL PROBABILITY (CONT.)
3.
Complement Rule:


The set of outcomes that are not in the event A is
called the complement of A, denoted AC.
The probability of an event occurring is 1 minus
the probability that it doesn’t occur: P(A) = 1 –
P(AC)
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FORMAL PROBABILITY (CONT.)
4.
Addition Rule (cont.):


For two disjoint events A and B, the probability
that one or the other occurs is the sum of the
probabilities of the two events.
P(A or B) = P(A) + P(B), provided that A and B are
disjoint.
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FORMAL PROBABILITY
5.
Multiplication Rule (cont.):


For two independent events A and B, the
probability that both A and B occur is the product
of the probabilities of the two events.
P(A and B) = P(A) x P(B), provided that A and B
are independent.
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PROBLEM 5


A consumer organization estimates that over a 1year period 16% of cars will need to be repaired
once, 7% will need repairs twice, and 1% will
require three or more repairs.
Suppose you own two cars.
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WHAT IS THE PROBABILITY THAT NEITHER
CAR WILL NEED REPAIRED THIS YEAR?
1.
2.
3.
4.
.76
.24
.5776
.0576
25%
25%
25%
25%
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2.
3.
4.
WHAT IS THE PROBABILITY THAT BOTH
CARS WILL NEED REPAIRED THIS YEAR?
1.
2.
3.
4.
.76
.24
.5776
.0576
25%
25%
25%
25%
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2.
3.
4.
WHAT IS THE PROBABILITY THAT AT LEAST
ONE CAR WILL NEED REPAIRED THIS YEAR?
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2.
3.
4.
.76
.24
.5776
.4224
25%
25%
25%
25%
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PROBLEM 9 

For a certain candy, 10% of the pieces are yellow,
20% are blue, 10% are green, and the rest are
brown.
If you pick a piece at random, calculate the
probability of the following…
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WHAT IS THE PROBABILITY THAT YOU PICK
A BROWN PIECE?
1.
2.
3.
4.
40%
60%
.4
.6
25%
25%
25%
25%
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3.
4.
WHAT IS THE PROBABILITY THE PIECE YOU
PICK IS YELLOW OR BLUE?
1.
2.
3.
4.
.1
.2
.8
.9
25%
25%
25%
25%
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3.
4.
WHAT IS THE PROBABILITY THE PIECE YOU
PICK IS NOT GREEN?
1.
2.
3.
4.
.1
.2
.8
.9
25%
25%
25%
25%
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2.
3.
4.
WHAT IS THE PROBABILITY THE PIECE YOU
PICK IS STRIPPED?
1.
2.
3.
4.
0
.1
.5
1
25%
25%
25%
25%
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PROBLEM 9

Now suppose you pick three pieces.

You observe three independent events.
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WHAT IS THE PROBABILITY OF PICKING THREE
BROWN CANDIES?
1.
2.
3.
4.
5.
.064
.4
.64
.936
1.2
20%
20%
20%
20%
20%
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2.
3.
4.
5.
WHAT IS THE PROBABILITY OF THIRD ONE
BEING THE FIRST RED?
1.
2.
3.
4.
0.008
0.128
0.2
0.8
25%
25%
25%
25%
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3.
4.
WHAT IS THE PROBABILITY THAT NONE
ARE YELLOW?
0.001
 0.729
 0.9
 0.999

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WHAT IS THE PROBABILITY OF AT LEAST
ONE GREEN CANDY?
0.1
 0.271
 0.729
 0.9

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PROBLEM 11

You roll a fair die three times

Again, each roll is independent of the last.

Sample space of one roll {1,2,3,4,5,6}
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WHAT’S THE PROBABILITY YOU ROLL ALL
5’S?
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2.
3.
4.
0.0046
0.1667
0.8333
0.9954
25%
25%
25%
25%
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4.
WHAT’S THE PROBABILITY YOU ROLL ALL
EVEN NUMBERS?
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3.
4.
0.125
0.5
0.875
1
25%
25%
25%
25%
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2.
3.
4
WHAT’S THE PROBABILITY THAT NONE ARE
DIVISIBLE BY 2?
1.
2.
3.
4.
0.125
0.5
0.875
1
25%
25%
25%
25%
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2.
3.
4
WHAT’S THE PROBABILITY THAT AT LEAST
ONE IS 3?
1.
2.
3.
4.
0.1667
0.4213
0.5787
0.8333
25%
1.
25%
25%
2.
3.
25%
4.
WHAT’S THE PROBABILITY THAT NOT ALL
ARE 3’S?
1.
2.
3.
4.
0.0046
0.1667
0.8333
0.9954
25%
25%
25%
25%
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4.
PROBLEM 15

On September 11, 2002, a particular state
lottery’s daily number came up 9-1-1. Assume
that no more than one digit is used to represent
the first nine months.
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WHAT IS THE PROBABILITY THAT THE WINNING
THREE NUMBERS MATCH THE DATE ON ANY
GIVEN DAY THAN CAN BE REPRESENTED BY A
THREE DIGIT NUMBER?
25%
25%
25%
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2.
3.
4.
25%
0.000
0.001
0.273
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4
WHAT’S THE PROBABILITY THAT THE WINNING
THREE NUMBER MATCH THE DATE ON ANY
GIVEN DAY THAT CAN BE REPRESENTED BY A 4DIGIT NUMBER?
25%
25%
25%
25%
1.
2.
3.
4.
0.000
0.001
0.273
1
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WHAT’S THE PROBABILITY THAT A WHOLE YEAR
PASSES WITHOUT THE LOTTERY NUMBER
MATCHING THE DAY?
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2.
3.
4.
0.001
0.694
0.761
0.999
25%
25%
25%
25%
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FOR NEXT WEEK

Monday HW4

Chapter 15 – More Probability Rules.
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