Chapter 7

1. List all the possible outcomes 2. Identify the outcomes that represent the same event and determine the probability of each event 3. Make a table listing each event and probability

Probability distribution

expected value to the casino of a particular bet

house edge

A roulette wheel has 38 umbers: 18 black numbers, 18 red numbers, and the numbers 0 and 00 in green. assume that all possible outcomes--the 38 numbers--have equal probability a. What is the probability of getting a red number on any spin?

18/38=0.474

A middle school principal needs to schedule six different classes-- alg, eng, his, span, science, and gym-- in six different periods. How many different class schedules are possible?

6!=^x5x4x3x2x1+720

If a committee of 3 people are needed out of a possible 8 candidates and there is not any distinction between committee members, how many possible committees would there be?

8!/(8-3)!x3!=8!/5!x3!=8x7x5x5!/5!x3!=8x7x6/3x2x1=56

If an international track event has 8 athletes partcipating and three medals (gold silver and broze) are to be awarded, how many different orderings of the top three athletes are possible?

8!/(8-3)=8x7x6

Consider families of two children. List all the possible outcomes for the birth order of boys and girls. If we are only interested in the total number of boys in the families, what are the possible outcomes of the event?

0, 1, 2

Consider families of two children. List all the possible outcomes for the birth order of boys and girls. If we are only interested in the total number of boys in the families, what are the possible events?

0B= GG 1B=GB, BG 2B=BB

Use the at least once rule to find the probability of at least one head when you toss three coins

1-(1/2)^3=1-(1/8)=7/8

Suppose you roll a single die. What is the probability of rolling either a 2 or a 3?

1/6+1/6=2/6

How many 7-number license plates are possible

10x10x10x10x10x10x10x10x10x10=10^7

Suppose an autmobile insurance company sells an insurance policy with an annual premium of $200. Based on data from past claims, the company has calculated the following probabilities: An average of 1 in 50 policyholders will file a claim of 2,000 An average of 1 in 20 policyholders will file a claim of 1000 An average of 1 in 10 policyholders will file a clam of 500 assuming that the policyholder could file any of the claims above, what is the expected calue to the company for each policy sold?

200+(-2000)x1/50+(-1000)x1/20+(-500)x(1/10=$60

What is the probability that in a standard shuffled deck of cards, you will draw a 5 or a spade

4/52+ 13/52-1/52=16/52=4/13

At least once rule is for what events?

Independent

most basic possible results of observations or experiments. For example, if you toss two coins, one possible outcome is HT and another possible outcome is TH

Outcome

The probability that dependent events A and B occur together is

P(A and B)=P(A)xP(B given A)

If A and B are independent events with probabilities P(A) and P(B):

P(A and B)=P(A)xP(B)

On a team of 10 swimmers, how many possible 4-person relay teams are there?

There are 10x9x8x7=5040

Find the theoretical probability of: a. Getting two heads when you toss two coins b. getting a 3 when you roll a six sided die

a. 1/2 b. 1/6

What is the probability of rolling three 6s in a row with a single die?

(1/6)(1/6)(1/6)=1/216

What is the probability that at least two people in a class of 25 have the same birthday?

(364/365)x(363/365)....=answer/365^24=1.348x10^61/3.126x10^61=0.431

If you draw one card at random from a standard deck, what is the probability that it is a spade?

13/52 or 1/4

A little league manager has 15 children on her team. How many ways can she form a 9-players batting order?

15!/6!=1,816,214,400

Two events are ______ if they cannot occur together, like the outcome of a coin loss, as shown to the right. For non-overlapping events A and B, the probability that either A or B occurs is shown below

P(A or B)= P(A)+P(B)

For overlapping events A and B, the probability that either A or B occurs is shown below

P(A or B)=P(A)+P(B)-P(A and B)

All selections come from a single groups of items

permutations

In ________ the order of arrangement matters

permutations

When do we use the permutations formula?

when order does matter

B. If patrons in a casino spin the wheel 100,000 times. How many times should you expect a red number?

The law of large numbers tells us that as the game is played more and more times, the proportion of times the wheel shows a red number should get closer to 0.474. In 100,000 tries, the wheel should come up red close to 47.4% of the time, or about 47,400 times

If you are interested only in the number of heads when tossing two coins, the possible events are 0 heads, 1 head, 2 heads. Suppose you repeat a two-coin toss 100 times and your results are as follows: -0 heads occurs 22 times -1 head occurs 51 times -2 heads occurs 27 times compare the relative freuqeuncy probabilities to the theoretical probabilities. Do you have reason to suspect that the coins are unfair?

The theoretical probabilities 0 heads: 1/4 or 0.25 1 head: 2/4 or 0.5 2 heads: 1/4= 0.25 Find the relative frquencies: 0 heads: 22/100=0.22 1 head: 51/100 2 heads: 27/100 They are fairly close. There is no reason to suspect the coins are unfair.

Identify the method that resulted in the following statements a. Im 100% certain that you'll be happy with this car b. Based on goverment data, the chance of dying in an automobile accident during a one-year period is about 1 in 8000 c. the probability of rolling a 7 with a 12-sided dice is 1/12

a. Subjective B. Relative c. theoretical

What type of problem is this: How many 7-number license plates are possible

arrangements with repetition

all selections come from a single group of items no items may be selected more than once and the order of arrangment does not matter

combinations

Consists on one or more outcomes that share a property of interest. For example, if you toss two coins and count the number of heads. The outcomes HT and TH both represent the same event of 1 head.

event

The __________ is the mistaken belief that a streak of bad luck makes a person "due" for a streak of good luck

gambler's fallacy

The __________________ applies to a process for which the probability of an event A is P(A) and the results of repeated trials are independent

law of large numbers

Two events are _____________ if they can occur together, like the outcome of picking a queen or a club, as shown to the right

overlapping

What type of event is this? What is the probability that in a standard shuffled deck of cards, you will draw a 5 or a spade

overlapping

What do you use to solve this:If an international track event has 8 athletes partcipating and three medals (gold silver and broze) are to be awarded, how many different orderings of the top three athletes are possible?

permutation formula

Which problem method do I pick to solve? A middle school principal needs to schedule six different classes-- alg, eng, his, span, science, and gym-- in six different periods. How many different class schedules are possible?

permutations

no item may be selected more than once

permutations

Approximating the probability of an event A by making many observations and counting the number of times event A occurs

relative frequency method (empirical method)

based on observations or experiments, is the relative frequency of the event of interest

relative frequency proability

A fact about _____ is For tossing a coin 6 times, the outcomes HTTHTH and HHHHHH are equally likely

streaks

an estimate based on experience or intuition

subjective probability

Suppose you are playing the coin toss fame, in which you win 1$ for heads and lose $1 for tails. After 100 tosses you are $10 in the hole because you flip perhaps 45 heads and 55 tails. the empirical probability is 0.45 for heads. so you keep playing the game. with 1000 tosses, you get 480 heads and 520 tails. does the result agree with the law of large numbers? Have you gained back any of your losses? explain

the proportion go heads in your first 100 tosses was 45%. After 1000 tosses, the proportion of heads has increased to 480 out of 1000, or 48%. This agrees with the law of large numbers, because the proportion grew closer to 50% However, after 1000 tosses, youve won 480, while losing 520, for a net loss of 40 In other words, your losses have increased from 10 to 40, despite the fact that the proportion of heads grew closer to 50%

1. Count the total number of possible outcomes 2. Among all the possible outcomes, count the number of ways the event of interest can occur 3. Determine the probability

theoretical method for equally likely outcomes

Based of the assumption that all outcomes are equally likely, is determined by dividing the number of ways an event can occur by the total number of possible outcomes

theoretical probability

You purchase 10 lottery tickets, for which the probability of winning some type of prize on a single ticket is 1 in 10. What is the probability that you will have at least 1 winning ticket among ten tickets?

1-(0.9)^10=0.651

The probability that at least two people in a class of 25 have the same birthday is

1-0.431=57%

Consider families of two children. List all the possible outcomes for the birth order of boys in the families, what are the possible events?

BB, BG, GG, GB

Assuming equal chance of having a boy or girl at birth, what is the probability of having two girls and two boys in a family of four children?

Possible outcomes (2 girls, 2 boys): GGBB GBGB BGBG BBGG GBBG BGGB Possible Outcomes (All girls): GGGG Possible outcomes (3 boys and 1 girl): GBBB BBBG BGBB BBGB Possible Outcomes (All Boys): BBBB Possible Outcomes (3 girls, 1 boy): GGGB GGBG GBGG BGGG 6/16

Two events are __________ if the outcome of one affects the probability of the other event.

Dependent

Is this dependent or independent? A three- person jury must be selected at random from a pool that has 6 men and 6 women. What is the probability of selecting an all-male jury?

Dependent (and probability)

the probability of an event, expressed as P(event), is always between 0and 1 (inclusive). a probability of 0 means the event is impossible and a probability of 1 means the event is certain.

Expressing probability

Two events are ___________ if the outcome of one does not affect the probability of the other event

Independent

If the process is repeated over many trials, the proportion of the trials in which event A occurs will be close to the probability (P)(a). The larger the number of trials, the closer the proportion should be to P(A)

Law of Large Numbers

A three- person jury must be selected at random from a pool that has 6 men and 6 women. What is the probability of selecting an all-male jury?

P(Juror 1 is male)=6/12 P(Juror 2 is male)=5/11 P(Juror 3 is male)= 4/10 P(3 men)=6/12x5/11x4/10=120/1320=0.091

If the probability of an event A is P(A), then the probability that event A does not occur is 1-P(A). Since the probability of a family of four children having two girls and two boys is 0.375, what is the probability of a family of four children not having two girls and two boys?

P(not 2 girls)= 1-0.375=0.625