Stats chapter 4
Define independent event?
When the probability of an event is not affected by a previous event.
In an experiment, a coin is flipped, and the coin can either land heads up, or tails up. What would we call the outcome where the coin lands heads up?
An event
In the context of a sample space, how do we distinguish an event from a subset?
Remember that events are subsets, and subsets are events.
Kasey (a girl) and Michael (a boy) are auditioning for a reality TV show about singing. The show's producers want to choose one boy and one girl. Including Kasey and Michael, there are 4 girls and 2 boys auditioning. Is the event of Kasey being selected independent from the event of Michael being selected, or are the events dependent? Also, what is the probability that the producers will select Kasey and Michael for the show?
Since the selection of Kasey doesn't affect whether or not Michael is selected, the two events are independent. Therefore, to find the probability of both being chosen, we multiply their respective probabilities together, yielding: 1/4 x 1/2 = 1/8 = 0.125.
In probability, a/an _____ is an event of a sample space.
Subset
Gary has a deck of 52 cards. He wants to know the probability of drawing the jack of spades and then drawing the two of hearts from the deck without replacing either card. What's the probability of this event? (A standard deck of 52 cards has 4 suits (hearts, clubs, spades, diamonds), each with 13 cards. Each suit has an ace, cards numbered 2 through 10, and a jack, a queen, and a king.)
The probability can be found by multiplying the probability of the first event (1/52) by the probability of the second event (1/51).
Kate and Kyle are playing a game. They must flip a coin and spin a spinner that has 12 equal sections numbered 1 through 12. What is the probability that Kyle will flip a heads and spin the spinner and get an even number?
The probability of getting heads, 1/2, must be multiplied by the probability of spinning an even number 6/12 (which reduces to 1/2). 1/2 x 1/2 = 1/4.
Sheri rolls a die eight times. She rolls a one three times, a two one time, a four one time, and a five three times. What is the relative frequency in this data set of rolling a four?
The total number of times the die is rolled is eight. She rolls a four one time. Therefore, the relative frequency is 1 in 8, 1/8, or 13%
Which of the following statements BEST explains how to determine whether two variables are independent?
Their relative frequency values are equal, or very close to equal.
Using a spinner with 12 equal sections numbered 1 through 12, what is the probability that Jim will spin a number less than 8?
Since there are 12 equal sections numbered 1 - 12, 7 out of the 12 sections are numbers less than 8. Therefore, the probability of spining a number less than 8 is 7/12.
If you roll a die three times, what is the probability of rolling only even numbers?
The numbers on a die are 1, 2, 3, 4, 5 and 6 Even numbers are: 2, 4 and 6 1st time P(even number) = 3/6, replaced 2nd time P(even number) = 3/6, replaced 3rd time P(even number) = 3/6 So that p(even number) 3 times = 3/6 x 3/6 x 3/6 = 27/216 = 1/8
Kyle works at a local music store. The store receives a shipment of new CDs in a box. In the shipment, there are 10 country CDs, 5 rock CDs, 12 hip hop CDs, and 3 jazz CDs. What is the probability that the first CD that Kyle pulls from the box will NOT be hip hop?
There are 18 CDs that are NOT hip hop out of the 30 total CDs. The ratio is 18/30 which reduces to 3/5.
If P(B|A) = P(B), then what do we know about events A and B?
They are independent events.
Kyle works at a local music store. The store receives a shipment of new CDs in a box. In the shipment, there are 10 country CDs, 5 rock CDs, 12 hip hop CDs, and 3 jazz CDs. What is the probability that Kyle will select a jazz CD from the box, and then, without replacing the CD, select a country CD?
1/29
Kyle works at a local music store. The store receives a shipment of new CDs in a box. In the shipment, there are 10 country CDs, 5 rock CDs, 12 hip hop CDs, and 3 jazz CDs. What is the probability that the first CD Kyle chooses from the box will be country?
1/3
Sandra has a bag of candies. There are twelve candies in total, and three of the candies are peanut butter. The packaging for each candy is the same, so Sandra doesn't know which candy she will pull from the bag. What is the probability that Sandra will pull a peanut butter candy from the bag?
25% or 1 in 4 chance
Calculate 4! (factorial)
4*3*2*1=24
Solve the expression 7P2 (P = permutation)
7P2 = 7! / (7 - 2)! = 5040 / 120 = 42 or 7 x 6 = 42
Jane is attempting to unlock her locker but has forgotten her locker combination. The lock uses 3 numbers and includes only the numbers 1 to 9. The digits cannot be repeated in the combination. How many possible locker combinations can be formed?
9P3 = 9!/(9 - 3)! = 362,880/720 = 504 Another way to look at it is there are 9 choices for the first number in the combination, 8 for the second number, and 7 for the third number: 9 x 8 x 7 = 504 possible locker combinations.
Solve the following combination:
A combination is an arrangement of objects where order does not matter. The formula for a combination is nCr = n!/(r!(n-r)!), where n represents the number of items and r represents the number of items being chosen at a time. in this case 5*2=10
Solve 8! (factorial)
A factorial is the product of all the positive integers equal to and less than your number. 8*7*6*5*4*3*2*1=40320
Solve 8! (the factorial of 8)
40,320
The probability of winning a prize at a ball toss at a carnival is 2/7. What is the probability of not winning a prize?
5/7
Solve the expression 5P4 (P = permutation )
5P4 = 5! / (5 - 4)! = 120 / 5! = 120 or 5 x 4 x 3 x 2 = 120.
Jimmy is making multi-flavored ice cream cones by scooping in different flavors one at a time. Jimmy has 6 different flavors but can only put 3 flavors in each cone. The order of the flavors is important to him as it affects how he tastes each ice cream. How many different arrangements of cones can Jimmy make?
6P3 = 6!/(6 - 3)! = 720/6 = 120 or 6 x 5 x 4 = 120.
There are 24 marbles in a bag. Eight of those marbles are blue, seven of the marbles are red, five of the marbles are green, and four of the marbles are white. What is the probability of pulling a white marble from the bag?
First, how many total marbles are there? There are 24 total. Next, what is the desired outcome? The desired outcome would be to pull a white marble from the bag. The total possible desired outcome is four because there are four white marbles. Therefore, our ratio would be: 4 in 24, or 1 in 6
Annie writes the numbers 1 through 10 on note cards. She flips the cards over so she cannot see the number and selects three cards from the stack. What is the probability that she has selected the cards numbered 1, 2, and 3?
For this problem, the probability for drawing 1, 2, or 3 for the first card is 3/10 since all three of the cards (1, 2, or 3) are in the stack and can possibly be drawn. For the second card, we assume success in drawing one of those numbers for the first card so there are two of the desired numbers left and a total of 9 cards (2/9). For the third card we assume success in drawing two of the desired cards in the first and second picks so only one of the desired cards is left in the 8 remaining cards. (1/8). Multiply these together: 3/10 ∗ 2/9 ∗ 1/8 = 6/720 = 1/120
There is a bag of red and blue marbles. If you keep grabbing marbles out of the bag without replacing them until you get a blue marble, is each grab an independent event? Why or why not?
No, because each time you take a marble out of the bag, it changes the probability of grabbing a blue marble.
Twelve employees at a local company decide to go out for lunch. Of the 12 employees, 8 are men and 4 are women. If a woman is the first to order, what is the probability that a man will be the next person to order lunch?
Number of employees = 12 P (first order is a woman) = 4/12 (total employees to place order is now 11) P (second order is a man) = 8/11
The probability of selecting a brown M&M from a bag is 1 out of 4. If 8 M&Ms are selected, what is the probability that at least one of them is brown? (The M&M is returned to the bag each time.)
P(selecting brown M&M) = 1/4 P(not selecting brown M&M) = 3/4 Since 8 selections are to be made = (0.75)^8 = 0.100 P(selecting 8 M&Ms and at least one of them being brown) = 1-0.100 = 0.9
On a 5 question, multiple-choice test, what is the probability that you will get at least one problem correct while guessing? Each question has 5 choices.
P(selecting correct answer) = 1/5 P(not selecting correct answer) = 4/5 Since there are 5 questions in the test = (4/5)^5 = 0.328 P(at least one problem correct) = 1-0.328 = 0.672
The probability of selecting a red M&M from a bag is 2/5. If 5 M&Ms are selected, what is the probability of at least one M&M being red?
P(selecting red M&M) = 2/5 P(not selecting red M&M) = 1-(2/5) = 3/5 Since 5 selections are to be made = 3/5 x 3/5 x 3/5 x 3/5 x 3/5 = 0.0778 P(selecting 5 M&Ms and at least one of them being red) = 1-0.0778 = 0.922
The name Joe is very common at a school and 1 out of every 10 students go by that name. If there are 15 students in one class, what is the probability that at least one of them is named Joe?
P(selecting the name Joe) = 1/10 P(not selecting the name Joe) = 9/10 Since 15 selections are to be made = (0.9)^15 = 0.206 P(at least one of 15 students is named Joe) = 1-0.206 = 0.794
What do you call the likelihood of a certain event occurring out of a total possible number of events?
Probability
Lisa has a two-sided coin with heads and tails. She also has a spinner with four colors: green, blue, red, and yellow. What is the probability of Lisa flipping the coin and getting heads and spinning the spinner to land on green?
The probabilities are 1/2 and 1/4. Multiply those to get the probability of 1/8, or 12.5%.
Jessie has a deck of 52 regular playing cards and a bag of six marbles. In the bag, there are two blue marbles, three green marbles, and one white marble. What is the probability of Jessie drawing an ace from the deck of cards and a blue marble from the bag? (A standard deck of 52 cards has 4 suits (hearts, clubs, spades, diamonds), each with 13 cards. Each suit has an ace, cards numbered 2 through 10, and a jack, a queen, and a king.)
The two probabilities are 4/52 and 2/6, so we multiply them to get 8/312, or 1/39.
The local bowling team plays in a 7-team league where each team plays every other team 4 times in a season. Using the combination formula, how many different games will be played in a season?
There are a total of 7 teams in the league. Each game requires 2 teams. There are, therefore, 7C2 ways of playing each other. This must happen four times since every team must play every other team four times. So the answer is 4 times 7C2 = 4 x (7 x 6 x 5!)/(2 x 1 x 5!) = 4 x (7 x 6 / 2) = 84.
Grace, Avery, and Noah are creating a maze for their pet gerbil, Sam. Grace bets that Sam will turn left when first entering the maze. Avery bets that Sam will turn right, and Noah bets that Sam will go straight. If there's an equal chance of Sam taking one of the given paths, then what is the probability that Grace or Noah will be correct?
These are mutually exclusive events. Therefore you will add 1/3 and 1/3 to get 2/3.
Mrs. Allison is preparing a cookies and milk party for her third grade class. There are 12 students that drink only whole milk, 8 students that drink only almond milk, 7 students that drink only skim milk, and 3 students that drink only soy milk. What is the probability that a student from Mrs. Allison's class drinks only almond or soy milk?
These events are mutually exclusive. Therefore, you can add the two events like this: 8/30 + 3/30 = 11/30
Melissa collects data on her college graduating class. She finds out that of her classmates, 60% are brunettes, 20% have blue eyes, and 5% are brunettes that have blue eyes. What is the probability that one of Melissa's classmates will be a brunette or have blue eyes?
These events are non-mutually exclusive. Therefore, you will first add the two probabilities .60 + .20 = .80 and then subtract the overlap .80 - .05 = .75, or 75%
Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that boys will place 1st, 2nd, and 3rd in the marathon?
To solve this problem, must look at the probabilities the the top three runners are boys and multiply them together, 15/20 x 14/19 x 13/18 = 91/228.
Twenty students compete in a school-wide marathon and each student is of comparable running ability. Of the 20 students, 15 were boys and 5 were girls. What is the probability that girls will place 1st, 2nd, and 3rd in the marathon?
To solve this problem, you must look at the probabilities the the top three runner are girls and multiply them together, 5/20 x 4/19 x 3/18 = 1/114.
While playing with a standard deck of playing cards, what is the probability that Jim's 5th card will be red after selecting 4 cards that were also red and not replacing them? (Hint: There are 52 cards in a deck which contains 26 red and 26 black cards.)
Total number of cards = 52 P (1st selection is Red) = 26/52 P (2nd selection is Red) = 25/51 P (3rd selection is Red) = 24/50 P (4th selection is Red) = 23/49 P (5th selection is Red) = 22/48 = 11/24
Using a standard deck of 52 cards, what is the probability of selecting a 4 and then after not replacing the card, selecting another 4?
Total number of cards in a deck = 52 Total number of cards with 4 = 4 So in the first selection p(4) = 4/52 If the card is NOT replaced, total cards remaining = 51 In the second selection p(4) = 3/51 So that the required probability = 4/52 x 3/51 = 12/2652 = 1/221
In a fish tank there are 10 clown fish, 4 angel fish, 6 puffer fish, and 5 eels. What is the probability of scooping out a clown fish on the second scoop after scooping out a clown fish on the first scoop and not replacing it? Give your answer as a fraction in its most reduced form.
Total number of fish at the beginning = 25 P (first selection to scoop out a clown fish) = 10/25, the clown fish is not replaced (number of scoop fish remaining = 9) P (second selection to scoop out a clown fish) = 9/24 = 3/8
The letters that spell out the state CALIFORNIA are cut and placed in a bag. What is the probability that the 3rd letter selected will be a C if the first two letters selected were both I's? (Letters were not replaced)
Total number of letters in CALIFORNIA is 10 P (first letter is I) = 2/10, so that 9 letters remain thereafter P (second letter is I) = 1/9, so that 8 letters remain thereafter P (third letter is C) = 1/8 (only 1 letter C in the bag present)
All of the letters that spell MISSISSIPPI are put into a bag. What is the probability of selecting a vowel, and then after replacing the letter, also drawing an S?
Total number of letters in the word = 11 Vowels : i ( 4 times i ) P(vowel) = 4/11 Since letter is replaced, total number of 11 letters still remain. P(obtaining S) = 4/11 So that P(vowel) and P(s) Replacing = 4/11 x 4/11 = 16/121
All of the letters of MISSISSIPPI are put in a bag. What is the probability of selecting both an M and then after not replacing the letter, selecting a P?
Total number of letters: 11 Number of M = 1 Probability of selecting M = 1/11 Since the card is NOT replaced, the total number of letters remaining = 10 Now Probability of selecting letter P = 2/10 (since there are 2 Ps in the word) Required probability = 1/11 x 2/10 = 2/110 = 1/55
Jenny has a bowl of M&M's that has 6 brown, 3 green, 4 red, and 12 yellow M&M's. She selects a yellow M&M and does not replace it. What is the probability that her second selection will be a brown M&M?
We are looking for one event, the possibility of selecting a brown M&M. If a yellow M&M is removed, the number of M&Ms remaining is 24 (out of which 6 are brown) P (selecting a brown M&M) = 6/24 = 1/4
Beatrice spins a spinner four times. Out of the four times, she lands on orange once. She tells Edward that she landed on orange 25% of the time. What is this an example of?
We can use this information to say that the spinner landed on orange 25% of the time which is an example of relative frequency.
When one event influences the outcome of another event in a probability scenario, we call that:
a dependent event
What is a set (S) of a random experiment that includes all possible outcomes of the experiment?
A sample space is a set (S) of a random experiment that includes all possible outcomes of the experiment
What do we call the probability of a second event given that a first event has already occurred?
Conditional probability
What is it called when the probability of an event is not affected by a previous event?
Independent probability
The probability of selecting a green ball from a bag is 1 out of 8, assuming there are 8 balls in the bag. If you select 3 balls, replacing them each time, what is the probability of at least one of them being green?
P(selecting a green ball) = 1/8 P(not selecting a green ball) = 7/8 Since 3 selections are to be made = 7/8 x 7/8 x 7/8 = 0.67 P(selecting 3 balls and at least one of them being green) = 1-0.67 = 0.33
What do we call the ratio of the occurrence of a singular event and the total number of outcomes?
Relative frequency is the ratio of the occurrence of a singular event and the total number of outcomes.
When we look at a two-way table, how are the values of joint relative frequency different from the values of marginal relative frequency?
Values of marginal relative frequency are found at the edges, while values of joint relative frequency are internal to these edges.
James is going through an old bag of marbles. He bets his friends that he can pull a red marble out of the bag and then a yellow marble in one try. There are 12 red marbles and 7 yellow marbles. What is the probability that he can pull a yellow from the bag, given that he has already pulled a red?
The answer will be 7/18 because there are 7 yellow marbles and James would have already removed one marble at this point, leaving 18 total marbles.
Steve has a regular deck of 52 playing cards. He wants to know the probability of pulling two clubs from the deck in a row without replacing the first club. What is the probability of this event? (A standard deck of 52 cards has 4 suits (hearts, clubs, spades, diamonds), each with 13 cards. Each suit has an ace, cards numbered 2 through 10, and a jack, a queen, and a king.)
To find the probability, multiply the probability of event A (13/52) by the probability of event B (12/51). 13/52 x 12/51 = 1/17.
When you conduct an experiment, what are you observing?
When you conduct an experiment, you are observing certain outcomes.
Why is the act of flipping a coin an example of independent probability?
Because the probability of a particular result of a coin flip is not affected by previous coins flips.
Using a standard deck of cards (which has 26 red cards and 26 black cards, with 13 cards of every suit), what is the probability of selecting a red card, and then after replacing the card, selecting a heart card?
Deck normally contains 52 cards Total number of red cards = 26 Number of Hearts = 13 P(Red card) and P(Heart) Replacing = 26/52 x 13/52 = 1/8
Sally is selling strawberry and regular lemonade at her stand. Out of all of the customers that bought her lemonade, 50% like regular lemonade and 30% like strawberry and regular lemonade. So if one of Sally's customers likes regular lemonade, what's the probability that they'll also like strawberry lemonade?
The correct formula is P(B|A)= P(A and B) / P(A) because you are finding the probability of B given A. P(A and B) = probablitiy that they like strawberry and regular lemonade = 30% P(A) = probability that they like regular lemonade = 50% P(B|A) = P(A and B) / P(A) = 30 / 50 = 0.6 = 60%
There is a bag of blue and red marbles and a deck of cards. What is the type of probability if one wants to know the chances of pulling out a blue marble from the bag and then an ace from the deck of cards?
The draw from the bag of marbles does not affect the draw from the deck of cards, there are independent events.
It's the first day of school and Anne is comparing her class schedule with her friends. Thirty percent of Anne's friends are in Geometry and World History with her. She has 60% of her friends in Geometry, and she has 40% of her friends in World History. What is the probability that one of her friends is in Geometry or World History with Anne?
This is a non-mutually exclusive probability. Therefore, you will first add the two probabilities .60 + .40 = 1, then subtract the overlap 1 - .30 = .70, or 70%.
Karen takes her group of third grade students out for ice cream. There is a total of 30 students. 13 of the students like chocolate ice cream, 12 of the students like strawberry ice cream, and 5 students like vanilla ice cream. When asked which two ice creams are their favorite, 8 students said they like both chocolate and strawberry ice cream. Out of the 30 students, what is the probability of a student liking ONLY chocolate or strawberry?
This is a non-mutually exclusive probability. Therefore, you will first add the two probabilities 13/30 + 12/30 = 25/30, then subtract the overlap: 25/30 - 8/30= 17/30
A locker combination contains four numbers between 1 and 20 and none of the numbers can be repeated. What is the probability that the locker combination will consist of all even numbers?
This problem involves finding two permutations: the number of total possible combinations and the number of possible combinations with only even numbers. The final probability is (even # combos/all possible combos), which reduces to the correct answer. The probabilities for [even numbers/all possible combinations] with no repetition gives us: (10 x 9 x 8 x 7) /(20 x 19 x 18 x 17) When you multiply these together and reduce the product you get 14/323.
Jimmy has the letters for the state of MISSISSIPPI written on cards, one letter per card. He turns the cards over and mixes up the order. If he selects one card at a time without replacing the cards, what is the probability that he will spell the word MISS in order?
To calculate the probability, we need to know the total outcomes and the favorable outcomes. The total outcomes is 11 x 10 x 9 x 8 because he only needs to select 4 cards and there are 11 cards and they are not being replaced. This equals 7,920. Favorable outcomes would be 1 x 4 x 4 x 3. For the first blank, there is only one M so a 1 goes in the first blank. The second blank, there are 4 I's, so a 4 goes in the 2nd blank. The 3rd blank, there are 4 S's and the 4th blank there are only 3 S's because one was already selected. This equals 48. Probability equals favorable outcomes over total outcomes. So, 48/7,920 equals 1/165.
