stats probability
The sample size is the
# of draws
A coin is tossed 10,000 times. The coin shows heads 4,850 times. Express the chance error both in absolute terms and as a percentage of the number of tosses.
-150, -1.5%
The probability that the parameter is in the interval is either _____ or ______.
0% or 100%
I buy 12 eggs from a local store. Unknown to me, 6 of the 12 eggs contain salmonella. I will choose 3 eggs at random from the 12 to make a cake. What is the chance that at least one of the 3 eggs contain salmonella?
1 - (6/12)(5/11)(4/10) = .9091 = 90.91%
A die is rolled four times. What is the chance that none of the rolls show 3 or more spots? Give your answer as a percent with two decimal places.
1.23
I have 25 light bulbs in a large box. Unknown to me, 5 of these 25 bulbs are broken. I select 3 of these bulbs at random from these 25 bulbs to put in a chandelier. What is the chance the third bulb I pick works? (You know nothing about the first two bulbs)
20/25
Three cards are dealt from a well-shuffled deck. Find the chance that none of the cards are diamonds. Give your answer as a percent rounded to 2 decimals.
41.35
mutually exclusive
A term we use to indicate two things where the occurrence of one of the things means the other thing cannot happen.
Frequency Theory
A theory used to define "chance" as it relates to chance processes that can be continuously repeated under the same conditions with the same possible outcomes for every repetition.
box model
A visual way to help model possible outcomes for a real-life chance process that may be repeated multiple times.
For a large number of die rolls, the percent of 6's is likely to be: A. Exactly 16.67% B. Close to, but not exactly 16.67% C. More than 20%
B. Close to, but not exactly 16.67%
Abraham de Moivre
One of the early founders of the formal study of the frequency theory of chance who wrote the book "The Doctrine of Chances."
Define Opposite Event.
The event that occurs if the given event does not.
A box contains 10,000 tickets: 4,000 0's and 6,000 1's. And 10,000 draws will be made at random with replacement from this box. Which of the following best describes the situation, and why?
The number of 1's is likely different from 6,000, but the difference is likely to be small compared to 10,000.
multiplication rule
The rule of chance that states, "The chance that both of two things happen is the chance the first will happen multiplied by the chance the second will happen given that the first happened."
dependent
The term used to describe two outcomes where the chance of each happening changes depending on whether or not the other has happened.
Select ALL of the following statements that are true concerning box models used to represent gambling games that are based on rolling a six-sided die.
The tickets in the box represent the possible outcomes for any single play. To represent playing the game many times in a row, tickets in the box ARE replaced after each play.
chance
This term is used to represent the percentage of time a specific result is expected to occur when the same basic procedure is repeated over and over again, where each repetition is independent.
the net gain
What the SUM of the draws represents for a box model that describes a gambling game based on chance where one can either lose or win money.
Unconditional Chance
When NO conditions are put on previous outcomes in the consideration of the chance of a later outcome.
unconventional chance
When NO conditions are put on previous outcomes in the consideration of the chance of a later outcome.
Chance error
When the actual outcome for a chance process does not match the expected outcome, and no bias is present.
chance error
When the actual outcome for a chance process does not match the expected outcome, and no bias is present.
Conditional Chance
When the chance of a later outcome puts a condition on a previous outcome.
conditional chance
When the chance of a later outcome puts a condition on a previous outcome.
A coin is tossed six times. Two possible sequences of results are (i) H T T H T H (ii) H H H H H H (The coin must land H or T in the order given; H=heads, T=tails.) Which of the following is correct?
i
The population size is..
the # of tickets in the box
The parameters describe
the percentage or avg. of the box.
The population is....
the tickets in the box
confidence intervals are used when..
you are reasoning backwards, from the draws to the box.
probabilities are used when..
you reason forward, from the box to the draws.
Probability =
# of ways desired outcome can occur/# of ways any outcome can occur
A pet store has 18 fish in a tank: 10 males and 8 females. I buy 2 of the 18 fish selected at random. What is the chance that one of my fish is male and the other is female?
(10/18)(8/17)+(8/18)(10/17) = .5229 = 52.29%
Three cards are dealt from a well-shuffled deck. Find the chance that all of the cards are diamonds. Give your answer as a percent rounded to 2 decimals.
1.29
Probabilist
A mathematician who specializes in computing probabilities of complex events.
addition rule
A rule of chance that states, "To find the chance that at least one of two things will happen, check to see if they are mutually exclusive, and if they are, add the chances."
Independent
The term used to describe two outcomes where each always maintains the same chance of happening, whether or not the other has happened.
The center of the histogram of the population is
avg of box
A pair of dice are thrown. Find the chance that both dice show 3 spots. Give your answer as a percent with 2 decimals.
2.78
Suppose that 20% of a certain farm's eggs contain salmonella. Are we more likely to get less than 15% in a sample of 20 or a sample of 100? Why?
20. The law of averages says that proportions far from the actual proportion are more common in smaller samples than in larger samples.
I have 25 light bulbs in a large box. Unknown to me, 5 of these 25 bulbs are broken. I select 3 of these bulbs at random from these 25 bulbs to put in a chandelier. What is the chance the first bulb I pick works?
20/25
A gambler will play roulette 50 times, betting a dollar on four joining numbers each time (like 23, 24, 26, 27 in figure 3, p. 282). If one of these four numbers comes up, she gets the dollar back, together with winnings of $8. If any other number comes up, she loses the dollar. So this bet pays 8 to 1, and there are 4 chances in 38 of winning. Her net gain in 50 plays is like the sum of draws from the box . Fill in the blanks.
50, a box with four $8's and thirty four $1's
A die will be rolled some number of times, and you win $1 if it shows an ace (one) more than 20% of the time. Which is better: 60 rolls, or 600 rolls? Explain.
60 rolls because you need a larger chance error.
A die is rolled four times. What is the chance that not all the rolls show 3 or more spots? Give your answer as a percent with two decimal places.
80.25 (with margin: 0.5)
Three cards are dealt from a well-shuffled deck. Find the chance that the cards are not all diamonds. Give your answer as a percent rounded to 2 decimals.
98.71
John Kerrich
A South African mathematician famous for carrying out experiments in probability theory with a coin while in a prisoner of war camp during World War II.
The Paradox of the Chevalier de Mere
A contradiction from the seventeenth century where the chance of obtaining at least one outcome on a die or a pair of dice was overestimated.
Consider the following two games: A. I roll a die 60 times. I win $100 if I roll at least eight 6's. B. I roll a die 600 times. I win $100 if I roll at least eighty 6's. Which game is better? Why?
A sample of 600. The law of averages says that proportions close to the actual proportion is more common in larger samples than in smaller samples.
According to genetic theory, there is very close to an even chance that both children in a two-child family will be of the same sex. Here are two possibilities: A. 20 couples have two children each. In exactly 10 of these families, it will turn out that both children are of the same sex. B. 50 couples have two children each. In exactly 25 of these families, it will turn out that both children are of the same sex. Which is more likely. Why?
A. Smaller sample sizes have better chance at getting exact probabilities.
A gambler will play roulette 30 times, betting a dollar on even each time. If an even number comes up, she gets the dollar back together with winnings of $1. If an odd number comes up (or 0 or 00), she loses the dollar. So this bet pays 1 to 1, and there are 18 in 38 chances of winning. Her net gain is like the sum of __________ draws from the box ____________.
Her net gain is like the sum of 30 draws from the box 18 $1 20-$1
How does the Frequency Theory view probability?
It looks at probability as the ratio of how many ways a certain outcome can occur compared to the number of ways it can either occur or not occur.
According the fictitious conversation between John Kerrich and the Assistant to the King of Denmark as recorded in section 1 of chapter 16, what specific reasoning does Kerrich use to help convince the Assistant that the chance of flipping 'tails' on a coin does NOT go up after a long string of 'heads' has occurred?
Kerrich explains that after each time he had flipped 4 'heads' in a row, about half of the time (69 out of 130) another 'head' appeared on the next flip, and about half the time (61 out of 130) a 'tail' appeared.
Explain the Gambler's Fallacy.
The notion that the longer a streak has held, the more likely the streak is to end.
Law of Averages
The term used to describe the phenomenon that as the number of repetitions of a chance process increases, the error between the expected outcome and actual outcome is likely to be large in absolute terms but small in relative terms.
The Law of Averages
The term used to describe the phenomenon that as the number of repetitions of a chance process increases, the error between the expected outcome and actual outcome is likely to be large in absolute terms but small in relative terms.
The histogram of the sample represents
each individual ticket that was drawn.
I buy 12 eggs from a local store. Unknown to me, 6 of the 12 eggs contain salmonella. I will choose 3 eggs at random from the 12 to make a cake. What is the chance that none of the 3 eggs contain salmonella?
(6/12)(5/11)(4/10) = .0909 = 9.09%
Two hundred draws will be made at random with replacement from a box with the following tickets. -3 -2 -1 0 1 2 3 If the sum of the 200 numbers drawn is -20, what is their average?
-0.1
Two hundred draws will be made at random with replacement from a box with the following tickets. -3 -2 -1 0 1 2 3 If the sum of the 200 numbers drawn is 30, what is their average?
.15
Frequency theory
A theory used to define "chance" as it relates to chance processes that can be continuously repeated under the same conditions with the same possible outcomes for every repetition.
Explain the Law of Averages.
In the long run, chance error will be large in absolute terms but small compared to the number of draws/plays/flips, etc
Define Independent.
Two things are independent if the chances for the second given the first are the same, no matter how the first one turns out. Pg. 230
Define Mutually Exclusive.
Two things are mutually exclusive when the occurrence of one prevents the occurrence of the other: one excludes the other. Pg. 241
Statistics describe..
either the sample percent or the sample average of the draws.
A deck of cards is shuffled and the top two cards are placed face down on a table. True or false: The chance of getting the ace of clubs and then the ace of diamonds is 1/52 x 1/52.
false
Four cards will be dealt off the top of a well-shuffled deck. There are two options: (i) To win $1 if the first card is a club and the second is a diamond and the third is a heart and the fourth is a spade. (ii) To win $1 if the four cards are of four different suites. Which option is better? Or are they the same?
ii
What the NUMBER of draws represents for a box model that describes a gambling game based on chance where one can either lose or win money.
the number of times the game is played