stats ch 5
A television channel conducts a study on the number of TVs per household in their service area. Out of 1000 households surveyed, 350 have one TV, 500 have two TVs, 120 have three TVs and 30 have four TVs. How many TVs does each household have on average?
1.83
Jon and Alex are playing an arcade game. Jon is pretty good at this game. He has an 80% chance at winning this game. He and Alex will both play 8 games. Jon will need to win 5 games to beat Alex. What is the probability that Jon will win exactly 5 games?
15%
Jon and Alex play a game where they each need to kick a ball into a net. They will each kick 11 balls, but Alex has a slight edge, with a 60% chance of scoring to Jon's 40%. What is the probability that Jon will score exactly 8 times?
2%
Sam is working on a computer programming project. He is creating a game that allows the player to shoot a ball into a basket in competition with the computer. The computer is set to make 60% of the shots on the medium level. What is the probability that the computer will make exactly 3 baskets out of 10 shots?
4% probability of success
The probability distribution for x is presented in the following table. What is the probability that the value of x will be equal to 2 and less than 4?
40%
Suppose a survey was conducted across the country regarding the number of firearms that people had in their households. If the sample size was 25,000 and 17,000 households reported that they had no firearms in their home, what would be the empirical probability that a randomly selected household had no firearms in their home?
68%
When the probability is a combination of the possible successes that are less than or equal to the trial number, to what are we referring?
Cumulative binomial distribution
What is the ratio of successful outcomes and the total number of trials?
Actual probability
Which of the following statements is TRUE in regards to a binomial experiment?
All choices are true.
Tom is playing a game of craps and and if the sum is 7 or 11, then he wins. If the sum is 2, 3 or 12, then he loses. If the sum is anything else, then he/she continues throwing until that number appears again, or he throws a 7, where the game ends in a loss. What would a binomial distribution of the data allow us to do?
Calculate the probability of what each toss will do.
In a poker game with a standard 52 card deck, what is the probability of getting a five-card hand with 4 aces?
1 in 54,145
What is the chance of rolling a 3 on any given six-sided die?
1 in 6
If an experiment has independent outcomes and a fixed number of trials, then what else will it need to be a binomial experiment?
Exactly two possible outcomes
Suppose you play a game with two 4-sided dice with sides numbered 1 through 4. If you roll a sum of 8 (face down), you win $10. If you roll anything else, you lose $1. What can you expect to win or lose in this game?
Lose 31 cents per game
Justin is conducting an experiment. He wants to know which type of pet is most preferred among 2nd graders. He asks them if they prefer dogs, cats, or hamsters. Is this a binomial experiment?
No, there are more than two possible outcomes.
Calculate the number of combinations of royal flushes
There are 4 combinations of royal flushes, one for each suit.
Calculate the number of combinations of getting a four-of-a-kind hand in poker.
There are 624 different combinations of four-of-a-kind hands in poker.
When one event affects the outcome of another event, we call them:
Dependent events
Jane is attempting to calculate the value of two consecutive coin tosses. What kind of data is she dealing with?
Discrete
What is data that cannot be divided, is distinct, and can only occur in certain values?
Discrete data
What is the probability related to discrete data?
Discrete probability
You conduct an experiment where you want to measure the number of rolls it takes to get two 6's in a row when you roll a fair six-sided die. State whether the random variable is discrete or continuous and give a summary of its values.
Discrete with values 2, 3, 4, 5, 6, etc.
You play a game with two six-sided dice. If you roll a sum of 6 or 8, you win $3. If you roll a sum of 11, you win $1, but for anything else, you lose $2. If you continue to play this game, what do you expect to win in the long run?
-$0.44 per roll
Suppose you play a game with two six-sided dice, where if you get doubles, you win $10, but for anything else you lose $5. If you continue to play this game, how much do you expect to win or lose per roll in the long run? In other words, what is the expected value of this game?
-$2.50
You're in a poker game with a standard 52 card deck. If you were to draw 3 consecutive cards, what is the probability of drawing three consecutive kings?
0.00018
In a poker game with a standard 52 card deck, what is the probability of drawing a five-card hand without any queens?
0.66
Vickie is playing a lottery where she will choose 3 numbers between 1 and 25. No numbers are repeated and the order of the numbers does not matter. What are her chances of picking the winning combination of numbers?
1 in 2,300
Naomi is buying a ticket for the lottery. She can pick 5 numbers, 1 through 15, with no numbers repeating. Order does not matter for her to win. What are her chances of winning?
1 in 3,003
Hannah is playing the lottery. She can pick 5 numbers between 1 and 32, with no numbers repeating. Order of these numbers does not matter. She can also pick a bonus number between 1 and 20. What are her chances of picking all of the numbers, including the bonus, correctly?
1 in 4,027,520
Hannah is playing the lottery. She can pick 5 numbers between 1 and 32, with no numbers repeating. The order of these numbers does not matter. She can also pick a bonus number between 1 and 20. What are her chances of picking all of the numbers, including the bonus, correctly?
1 in 4,027,520
24 In a poker game with a standard 52 card deck, what is the probability of getting a five-card hand with 4 jacks and a king?
1 in 649,740
When you roll two six-sided dice, what is the probability of getting a sum of 11?
1/18
When two six-sided dice are rolled, what is the probability of getting doubles? (two ones, two twos, etc.)
1/6
Alysha is at a local library with her best friend. They've noticed a lot of cute boys in the library, and Alysha bets that of the next twenty people to enter the library, exactly 12 will be cute boys. Assuming there is a 50% chance that the next person to walk into the library is a cute boy, what is the probability that Alysha will win the bet?
12%
Jon and Alex play a game where they each need to shoot a puck into a net. They are equally matched (thus each has a 50% chance of getting the puck in the net) and will each shoot 12 pucks. What is the probability that Jon will score exactly 7 times?
19%
Jon and Alex decide to play skee ball. This game only has one ring, and the guys must roll the ball into the 10,000 point ring or they won't get any points. Each of them has a 50% chance of success with each roll and will each roll 6 balls. What is the probability that Jon will get the ball in the ring exactly 4 times?
23%
A study has shown that there is a 40% chance that people who are 25 will still be alive when they are 85 years old. The researcher had randomly selected 3 college students aged 25. By using the binomial probability formula, what is the probability that the number alive at age 85 will be exactly two?
28.8%
The probability distribution for x is presented in the following table. What is the expected value of the random variable x?
3.1
Suppose you toss three coins. What is the probability that you get two heads and one tail if the order in which you get them does not matter?
3/8
Tom is playing a game that has a probability of success, P, of 0.6. He plays 3 games (n = 3). Using the binomial probability formula, what is the probability that he will win 2 games (x =2)?
43.2%
Say you're playing Blackjack, and you have a four and a jack. If you see that one of the dealer's two cards is a five, then what's the probability that you will be dealt a card that helps you?
54%
In a game of blackjack, the first card you draw is a nine and the second card is a seven. What is the probability that if you draw another card that you will lose (go over 21)? Assume that aces can count as 1 and you are playing with a standard 52 card deck.
60%
What is the binomial probability formula?
B(x; n, P) = C * P^x * (1 - P)^(n - x)
Suppose you play a game where you spin a spinner (see picture below) with areas of the colors on the spinner broken down as shown: 10% blue, 60% green, and 30% red. In addition, if the spinner lands on red you win 6 points, if it lands on blue you win 1 point, and if it lands on green you lose 5 points. If you keep spinning, how many points can you expect to win or lose per game?
I expect to lose 1.1 points per game.
You play a game where you toss a coin. On each toss if it lands with heads up, you win $1. However, if it lands with tails up, you lose $2. If you continue to play this game, how much can you expect to win or lose per game?
I expect to lose 50 cents per game.
What is the degree in which the variables are different from the mean called?
Standard deviation
What are expected values?
The results we can anticipate given that a particular event occurs.
What is the ratio of the desired outcome and the total number of possible outcomes?
Theoretical probability
What is the formula for the standard deviation of a binomial random variable?
sqrt (n * P * (1-P))
Suppose you play a game where you roll 2 six-sided dice. If you roll a 7, you will lose some money, but if you roll anything else you will win $1. How much money would you have to lose in order for you average $0 per roll?
$5
What is game theory?
A field of math used to guide logical decision making
Using the table below, find the probability of 1 success out of 10 trials, with a probability of .5.
Approximately 1%
What is an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure?
Binomial experiment
Which of the following assigns probabilities to the potential values of the variable x?
Binomial formula
What is the formula for binomial combinations?
C = n!/{(n - x)!x!}
Calculate the probability of getting a straight in a hand of poker.
There is approximately a 1 in 255 chance of getting a straight in poker.
A game is played where you draw two cards from a deck of 52 cards. If the cards are both jacks, then you win $5. Otherwise you lose $1. If you play this game, then what can you expect to win or lose in the long run?
lose $0.97 per game
What variable represents the number of trials in an experiment?
n
Which variable represents the number of trials when solving the expected value formula?
n
Assuming that an ace counts as a one, calculate the number of combinations of straight flushes for a poker hand. (In other words, disregard the royal flush.)
There are 36 possible combinations of straight flushes.
How many different combinations of flushes are there in a fair deck of cards?
There are 5,108 combinations of flushes in a deck.
Using the table below, find the probability that there are 3 successes out of 10 trials, with a probability of .2.
approximately 88%
What do we call the number of successful outcomes expected in an experiment?
Expected value
What is the number of successful outcomes expected in an experiment called?
Expected value
Why would a person who has few wrecks benefit from a lower premium?
Because even though a low premium comes with a high deductible, if the driver has a low chance of a wreck, they're less likely to ever need to pay the deductible.
Tom is playing a game of craps and if the sum is 7 or 11, then he wins. If the sum is 2, 3 or 12, then he loses. If the sum is anything else, then he continues throwing until that number appears again or he throws a 7, where the game ends in a loss. If the probability of Tom winning is 0.4929, why is it not profitable for him to play the game over the long-term?
Because his probability of winning is lower than 50%.
What is the number of successes in a binomial experiment called?
Binomial Random Variable
Identify in which of the following cases is the probability 11/221:
If, in a game of blackjack, you were to draw 2 face cards (king, queen, or jack) in a row.
Which of the following statements BEST exemplifies discrete data?
Jim flips a coin and it lands on heads 8 times.
At a particular casino, you pay $1 to play a hand of Blackjack. If you get a total of 21 in your first two cards, then you win $10. If not, you lose your dollar. If you are the only one playing this game, then how much do you expect to win or lose per hand?
Lose $0.52 per hand
Ally is trying to determine if an experiment is binomial. She is asking twenty people in front of a pet store if dogs' mouths are cleaner than a toilet. She asks the participants to answer true or false to her question. As far as she knows, the responses of the previous participants do not influence the responses of the other participants. Could this be considered binomial?
Yes, there are two possible outcomes, the outcomes are independent, and there is a fixed number of trials.
What is a probability formula that uses factorials to find the number of possible combinations of all the outcomes in the experiment?
combination formula
What do we call data that cannot be divided, which is distinct, and can only occur in certain values?
discrete data
The _____ is the mean of the distribution.
expected value
From mortality tables it has been determined that the probability of a 20-year old male non-smoker dying within the year is 0.0035. Suppose an insurance company wants to sell a $50,000 1-year life insurance policy to a 20-year old non-smoking male. What should they charge for the policy to break even?
$175
Suppose you draw two cards from a standard deck of 52. What is the probability that they are both kings?
1/221
A study has shown that there is a 40% chance that people who are 25 will still be alive when they are 85 years old. The researcher had randomly selected 3 college students aged 25. By using the binomial probability formula, what is the probability that the number alive at age 85 is two or three?
35.2%
Jon and Alex play a game where they shoot water at a target in order to make a car move on a track. The person that gets their car over the finish line first wins. Alex and Jon decide to play the game 3 times. If they are evenly matched, then what is the probability that Jon will win 2 games?
38%
Calculate the probability of getting a single pair in a hand of poker. Round that probability to the nearest percent.
42%
A study has shown that there is a 40% chance that people who are 25 will still be alive when they are 85 years old. The researcher had randomly selected 3 college students aged 25. By using the binomial probability formula, what is the probability that the number alive at age 85 is at least 1?
44.4%
Asia is conducting an experiment for her psychology class. She asks 80 students if they believe in hypnosis. Previous studies show that 60% of the population believes in hypnosis. Based on the expected value, how many people should answer yes, they do believe in hypnosis, in Asia's study?
48 people
Suppose a survey was conducted across the country regarding the number of firearms that people had in their households. Let X = the number of firearms in a household. From the survey of 30,000 households it was determined that the empirical probability of X = 1 was 0.2. How many of the households in the survey had one firearm in their home?
6,000
The probability distribution for x is presented in the following table. What is the probability that x is greater than or equal to 3?
60%
In a game of blackjack where you are the only player, the first card you draw is the king and the second is an eight. You decide to draw another card. What is the probability that upon drawing another card, you will lose (go over 21)? Assume that aces can count as 1 and you are playing with a standard 52 card deck.
76%
A game of Blackjack is played. Suppose you are the first player dealt with a new pack of cards. What is the probability you get an ace and a jack (in any order)?
8/663
A roulette wheel has 38 slots labeled with the numbers 1 through 36 and then 0 and 00. Slots 1 through 36 are colored either red or black. There are 18 red and 18 black. Slots 0 and 00 are colored green (see picture). On one spin of the roulette wheel, what is the probability that the ball lands on a red slot?
9/19
Which of the following is NOT a property of a random variable?
A random variable cannot be negative.
If the probability of a customer entering a store and purchasing a spoon tie is 35%, then what is the expected value of customers purchasing a spoon tie if 79 enter the store?
Approximately 28 customers
What is the expected value of 79 customers purchasing a spoon tie if the success probability is 35%?
Approximately 28 customers
What do we call the probability of each success in a given binomial experiment?
Binomial distribution
What do we call an experiment that contains a fixed number of trials that results in only one of two outcomes: success or failure?
Binomial experiment
What is an experiment that contains a fixed number of trials that results in only one of two independent outcomes: success or failure?
Binomial experiment
What is a probability formula that uses factorials to find the number of possible combinations of all the outcomes in an experiment?
Combination formula
What variable represents the probability of success on an individual trial?
p
After finding the number of ways to get four-of-a-kind for four cards in a poker hand, should you use: a. 48 choose 1 or b. 12 choose 1 x 4 choose 1 to describe all possible ways for getting the 5th card? Explain your reasoning.
Either way is correct, because both calculations describe all possible ways for getting the 5th card.
Children born after the turn of the century have a 60% probability of needing braces. What are the expected value and standard deviation for a group of 30 children surveyed?
Expected Value: 18 Standard Deviation: 2.7
A roulette wheel has 38 slots labeled with the numbers 1 through 36 and then 0 and 00. Slots 1 through 36 are colored either red or black. There are 18 red and 18 black. Slots 0 and 00 are colored green (see picture). Suppose that for a bet of $1 on red, the casino will pay you $2 if the ball lands on a red slot (a net gain of $1), and otherwise you lose your dollar. What can you expect to win or lose in this game?
I expect to lose about 5 cents per game.
A binomial experiment must have two possible outcomes: success and failure. Select the answer that is an example of a binomial experiment.
Picking an ace (success) or not an ace (failure) out of a deck of cards. Replacing each of the cards before drawing again. Drawing 20 cards total.
Jeanette asks 20 people in her history class if they are comfortable using technology. Seven out of the 20 say they are comfortable using technology. Calculate the theoretical and the actual probability of this scenario.
Theoretical: 50% Actual: 35%
Joseph asks 20 people in his math class if they are comfortable with using technology. Seven out of the 20 say they are comfortable with using technology. Calculate the theoretical and the actual probability of this scenario.
Theoretical: 50% Actual: 35%
Jeanette asks 50 students in her math class if they are comfortable using technology. Thirty-eight say they are comfortable using technology. Calculate the theoretical and actual probability of this scenario.
Theoretical: 50% Actual: 76%
Calculate the number of combinations and the probability of getting a three-of-a-kind hand in poker.
There are 54,912 different combinations of three-of-a-kind hands in poker. There is approximately a 1 in 47 chance of getting a three-of-a-kind hand in poker.
You decide to conduct a survey of families with two children. You are interested in counting the number of boys (out of 2 children) in each family. Is this a random variable, and if it is, what are all its possible values?
Yes, it is a random variable, and its values are 0, 1, or 2.
Suppose you play a game where you roll 2 six-sided dice. If you roll a 4, 5, or 6, you will lose $6, but if you roll anything else, then you win $3. How much money will you gain or lose per each throwing?
You will average exactly 0 per each throwing of the dice.
20 Suppose you play a game where you roll 2 six-sided dice. If you roll a 7, you gain $108, but if you roll anything else, you lose $12. How much money will you gain or lose per each throwing?
You will, on average, earn $8 per each throwing of the dice.
An agency decides to conduct a survey on household incomes in their county. Let x = the household income. What type of variable is x?
x is a continuous random variable.
You decide to collect a bunch of cans of soda and measure the volume of soda in each can. Let x = the number of mL of soda in each can. What type of variable is x?
x is a continuous random variable.