Chapter 4Probability and Sampling Distributions

PigA probability game

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 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 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?

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: 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:

Event – any collection of outcomes from the sample space of a chance experiment.

Probability of an event E 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: 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 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 Probability1. 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 = 1/133. 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

0 721 14242 26193 32274 15455 10236 762

Find the probability of a person having:a. Two televisions

2619/10672b. Between one and three televisions, inclusive

7270/10672 = 3635/5336c. Seven televisions

0

Probability with addition rule

To find the probability of one event or the other to occur, add the probabilities of each one happening

1. 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 = 1/62. 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 If two events are independent events then the

probability of both things happening is P(A) · P(B)1. 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) = .617782. From the problem above find the probability that

none of the applications are approved.(.15)(.08)(.21) = .00252

Ways of finding # of outcomes: Counting rule Permutations – in a line Arrangement in line with

duplicates Arrangement in circle Combinations – in a group

Counting rule 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.

1. 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 = 122. 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 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 Arrangement of distinct objects in a particular order Equation: n! or nPr or

1. 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! = 50402. 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 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

)!(!rnn

The number of different permutations of n things of which p are alike, q are alike, or r are alike Equation: How many different ways can we

arrange the letters in the word “STATISTICS”.

!!!!rqp

n

Arrangement of distinct objects in a circle Equation: (n – 1)! How many different ways can four

people be arranged in a circle?3! = 6

Combinations A selection from a collection of distinct objects where order

is not important. Equation: nCr or 1. 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 = 437582. 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?

10C4 = 210

)!(!!rnr

n

Odds 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.

1. Find the odds against rolling a 5 when a single die is rolled once.

5 : 12. 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 : 203. What are the odds against winning?

20 : 18