Download Chapter 4 Probability and Sampling Distributions

Chapter 4
Probability and Sampling
Distributions
Pig
A probability game
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Toss the die and record the number
Toss the die again, write down the number and add it to the
previous number
Players may continue to play and accumulate points or they
may decide to stop at any point before the die is thrown again.
If stopped the student may not collect any more points but get
to keep all points earned
When a one is tossed, every student still playing loses all of
his/her points for that round and the round is over
A game is three rounds. Highest point total wins the game
The Cherrios Experiment
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Many companies are putting toys in their products to try to get
customers to buy more. The company that makes Cheerios
thinks this might be a good way to get families to buy more
boxes of Cheerios. They will make six different toys and put one
in each box of Cheerios and Multi-Grain Cheerios. That way kids
will want their parents to keep buying Cheerios until they have
all six different toys.
How many boxes do you think a family will need to buy in order
to obtain all six toys?
Perform a simulation with a die to predict how many boxes of
cereal a family will need to buy to be sure they collect all 6 toys.
Each day decisions are based
on uncertainty
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Should you buy an extended warranty
for your new digital camera?
Should you allow 25 min to get school
or is 15 min enough?
If an artificial heart has four key parts,
how likely is each one to fail? How
likely is it that at least one will fail?
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Chance experiment – any activity or situation
in which there is uncertainty about which of
two or more possible outcomes will result.
Chance behavior is unpredictable in the short
run but has a regular and predictable pattern
in the long run
Sample space – the collection of all possible
outcomes of a chance experiment.
Sample space of chance
experiment:
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Experiment - whether men and
women have different shopping
preferences when buying a CD
at a music store – classical,
rock, country, and “other”.
Tree diagram of sample space:
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Event – any collection of outcomes from
the sample space of a chance
experiment.
Probability of an event E
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P(E) = number of outcomes favorable to E
number of outcomes in the sample space
***only when the outcomes of an experiment are
equally likely – fair coins or dice, etc.
Probability rules:
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Any probability is a number between 0
and 1
All possible outcomes together must
have a probability of 1
The probability that an event does not
occur is 1 minus the probability that the
event does occur
Probability Model
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A mathematical description of a random
phenomenon consisting of a sample
space and a way of assigning
probabilities to events.
Construct a probability model for a
family with 3 children.
All probabilities must add to one
Pg 226 #4.19, 4.20, 4.21
Basic Probability
1.
What is the probability of rolling a 3 with one die?
1/6
2.
What is the probability of picking a Queen in a deck
of cards?
4/52 = 1/13
3.
What is the probability of picking a heart in a deck
of cards?
1/4
4.
What is the probability of rolling a sum of 8 with
two dice?
(2,6)(6,2)(3,5)(5,3)(4,4) = 5/36
5.
What is the probability of a family with three children to have all
girls?
1/8
6.
A survey was taken of 10,672 families to determine the number
of televisions owned by each. The following results were
obtained:
Number of Televisions Owned
Frequency
0
1
2
3
4
5
6
Find the probability of a person having:
a.
Two televisions
2619/10672
b.
Between one and three televisions, inclusive
7270/10672 = 3635/5336
c.
Seven televisions
0
72
1424
2619
3227
1545
1023
762
Probability with addition
rule
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1.
To find the probability of one event or the other to
occur, add the probabilities of each one happening
Two dice are rolled. What is the probability that the
sum of the dots appearing on both dice together is 9 or
11?
4/36 + 2/36 = 6/36 = 1/6
2.
One card is drawn from a deck of cards. What is the
probability of getting a king or a red card?
4/52 + 26/52 – 2/52 = 28/52
Multiplication rule
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1.
If two events are independent events then the
probability of both things happening is P(A) · P(B)
Philip, Janet, and Fredric have each applied to different
banks for a home equity loan. The probability that
Philip’s application is approved is .85. The probability
that Janet’s application is approved is .92, and the
probability that Fredric’s application is approved is .79.
Assuming independence, find the probability that all
three applications are approved.
(.85)(.92)(.79) = .61778
2.
From the problem above find the probability that none
of the applications are approved.
(.15)(.08)(.21) = .00252
Ways of finding # of
outcomes:
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Counting rule
Permutations – in a line
Arrangement in line with duplicates
Arrangement in circle
Combinations – in a group
Counting rule
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1.
If one thing can be done in m ways and if after this is done,
something else con be done in n ways, then both things can be
done in a total of (m)(n) different ways in the stated order.
A certain model car comes with one of three possible engine sizes
and with or without an AM/FM radio. Furthermore, it is equipped
with automatic or standard transmission. In how many different
ways can a buyer select a car?
3 · 2 · 2 = 12
2.
There are nine approach roads leading to an airport. Because of
heavy traffic, a taxi driver decides to go to the airport by one road
and to leave by another road. In how many different ways can this
be done?
9 · 8 = 72
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How many different numbers greater than 3000 can be formed from
the digits 2, 3, 5, and 9 if no repetitions are allowed?
3 · 3 · 2 · 1 = 18
Permutations
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1.
Arrangement of distinct objects in a particular order
Equation: n! or nPr or n!
(n  r )!
In a supermarket there is a long line at the checkout counter. The
manager notices this and decides to open an additional checkout
counter. Seven people rush over to the new checkout counter. In
how many different ways can these seven people line up to be
checked out?
7! = 5040
2.
Susan is an IRS agent. She has made appointments with eight
taxpayers to review their 1040 tax forms on May 2 on a first-come,
first-served basis. However, due to a computer malfunction, she
finds that she has time to meet with only five taxpayers to review
their forms. Assuming order counts, in how many different ways
can this be done?
8P5
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or 8 · 7 · 6 · 5 · 4 = 6720
Each year movie-goers in a certain city are asked to rank the 5 best
movies from among a list of 14 movies. In how many different ways
can this be done?
14P5
or 14 · 13 · 12 · 11 · 10 = 240240
The number of different permutations of n
things of which p are alike, q are alike, or r are
alike
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n!
p!q!r!
Equation:
How many different ways can we
arrange the letters in the word
“STATISTICS”.
Arrangement of distinct
objects in a circle
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Equation: (n – 1)!
How many different ways can four
people be arranged in a circle?
3! = 6
Combinations
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1.
2.
A selection from a collection of distinct objects where order is not
important.
n!
Equation: nCr or
r!(n  r )!
Medical researchers are testing a new drug for treating one form of
a neurological disorder. It is decided to select a random group of
18 people and then to select 8 of these people to be given the new
drug. The remaining 10 people will be given a placebo. In how
many different ways can the 8 subjects be selected?
18C8
= 43758
10C4
= 210
There are ten nurses who work on the night shift on the tenth floor
of General Hospital. In an effort to save money, the hospital
administrator decides to fire four nurses. In how many different
ways can the administrator select the four nurses to be fired?
Odds
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1.
The odds in favor of an event occurring are p to q,
where p is the number of favorable outcomes and
q is the number of unfavorable outcomes.
Find the odds against rolling a 5 when a single die
is rolled once.
5:1
2.
A roulette wheel has 38 slots. One slot is 1,
another is 00, and the others are numbered 1
through 36, respectfully. You are placing a bet that
the outcome is an odd number. What are the odds
of winning?
18 : 20
3.
What are the odds against winning?
20 : 18