Independent, Dependent, Mutually Exclusive, and Overlapping Events
Expected Value
Counting
Summary Statistics
100
If you have a bowl of crayons, 5 red, 3 green, and 2 blue, this is the probability of choosing a red crayon.
What is 5/10 = 1/2?
100
If you have a bowl of crayons, 5 red, 3 green, and 2 blue, this is the probability of choosing a red crayon or a blue crayon.
What is 7/10?
This is an mutually exclusive problem. We know we are going to be adding because of the word OR.
The probability of choosing red is 5/10.
The probability of choosing blue is 2/10.
So the probability of one or the other is 5/10+2/10 = 7/10.
100
You have a fair spinner. One half of the spinner is worth 2 points. The other is worth -1 points. This is the expected value of spinning the spinner.
What is 0.5 points?
Your expected value is taking your possible points, multiplying them by their probabilities, then adding the products.
You have a 1/2 chance of winning 2 points
You have a 1/2 chance of losing -1 points
EV = 2(1/2) + (-1)(1/2) = 1/2 = 0.5
100
You go to a clothing store and find two shirts, three pairs of pants, and a hat that you like. You may only buy one item. This is how many things you have to choose from.
What is 6?
When you have items to choose that you are not combining into something new, then you using the adding counting principle.
You have 2 shirts + 3 pants + 1 hat = 6 items
100
This is the median of the following numbers:
2, 6, 9, 5, 7, 8, 8, 4
What is 6.5?
First, put the numbers in order:
2, 4, 5, 6, 7, 8, 8, 9
There are two medians, 6 and 7, so we find their mean.
(6+7)/2 = 13/2 = 6.5
200
You have a die. This is the probability you roll an even number.
What is 1/2?
You have 6 possible outcomes, and 2, 4, and 6 are even, so you have 3/6 chances of getting an even = 1/2
200
You are playing with a deck of cards, and randomly choose one from the deck. This is the probability you get a face card (Jack, Queen, or King) or a red card.
What is 8/13?
This is an overlapping problem. We know we are going to be adding because of the word OR. And we know it is overlapping because there are cards that are red AND Jack, Queen or King.
So we are going to add the two probabilities, then subtract the probability of the cards that are BOTH red and face.
P(red) = 26/52 (half the cards are red - diamonds and hearts)
P(face) = 12/52 (there are 3 face cards in each suit, and four suits)
P(faceANDred) = 6/52 (half of the face cards are red)
26/52+12/52-6/52 = 32/52 = 8/13
200
You have a fair spinner split into equal thirds. One part of the spinner is worth 6 points. Another part is worth 3 points. The last part is worth -12 points. This is the expected value of spinning the spinner.
What is -1 points?
Your expected value is taking your possible points, multiplying them by their probabilities, then adding the products.
You have a 1/3 chance of winning 6 points
You have a 1/3 chance of winning 3 points
You have a 1/3 chance of losing -12 points
EV = 6(1/3) + 3(1/3) + (-12)(1/3) = 2+1+(-4) = -1
200
You go to a Tex Mex restaurant to order a taco. You can choose a soft taco or hard taco; beef, chicken, or tofu; and two different kinds of cheeses. This is how many different tacos you can order.
What is 12?
You are combining ingredients into something new, so you need to use the multiplication counting principle.
2 types of tacos x 3 meats x 2 cheeses = 2x3x2 = 12
200
This is the mean of the following numbers:
2, 6, 9, 5, 7, 8, 8, 4
What is 6.125?
We add the numbers, and divide by how many there are.
2+6+9+5+7+8+8+4 = 49
49/8 = 6 and 1/8 = 6.125
300
You walk into a friend's house that you are not familiar with. In the dim light you can see 11 different light switches. 2 will turn on the lights in the living room. This is the probability that you choose a light switch that turns on the lights in the living room.
What is 2/11?
300
You flip a coin and roll a die. This is the probability that you flip heads and roll a number divisible by 3.
What is 1/6?
This is an independent problem. We know we are multiplying because of the use of the word AND.
The probability of flipping heads is 1/2
The probability of getting a number divisible by 3 is 2/6 (there are 6 outcomes, and 3 and 6 are divisible by 3)= 1/3
So 1/2 x 1/3 = 1/6
300
You have a fair spinner split into equal fifths. One part of the spinner is worth 25 points. Another part is worth 100 points. Another part is worth 200 points. Another part is worth 0 points. The last part is worth -200 points. This is the expected value of spinning the spinner.
What is 25 points?
Your expected value is taking your possible points, multiplying them by their probabilities, then adding the products.
You have a 1/5 chance of winning 25 points
You have a 1/5 chance of winning 100 points
You have a 1/5 chance of winning 200 points
You have a 1/5 chance of winning 0 points
You have a 1/5 chance of losing -200 points
EV=25(1/5)+100(1/5)+200(1/5)+0(1/5)+(-200)(1/5)=
5+20+40+0+(-40) = 25
300
This number is equal to 4!
What is 24?
4! = 4x3x2x1 = 24
300
This is the range of the following numbers:
2, 6, 9, 5, 7, 8, 8, 4
What is 7?
First, put the numbers in order:
2, 4, 5, 6, 7, 8, 8, 9
Then we take the max - min
9-2 = 7
400
You have a bag of skittles with 10 red, 20 yellow, 15, green, 13 purple, and 7 orange skittles. Your friend borrows the bag and eats all of the reds. You get the bag back and randomly pick a skittle. This is the probability you choose a purple skittle.
What is 13/55?
You started with 65 skittles, but your friend ate 10 of them, so you only have 55 left. 13 of those are purple. So the probability is 13/55.
400
You have a plate of cookies with 5 chocolate chip, 5 snickerdoodles, 4 sugar, and 6 oatmeal raisin cookies. This is the probability of randomly choosing a sugar cookie, eating it, and then randomly choosing a chocolate chip cookie.
What is 1/19?
This is a dependent problem because it changes when we remove the sugar cookie. We know we are going to multiply because of the words AND THEN.
The probability of getting the sugar cookie is 4/20 = 1/5
After you remove it, there are only 19 cookies left, so the probability of getting a chocolate chip after is 5/19
1/5 x 5/19 = 5/95 = 1/19
400
The first prize for a raffle is $5,000 (with a probability of 0.001) and the second prize is $2,000 (with a probability of 0.003). Otherwise you don't win anything. This is the expected value of playing this game.
What is $8?
Your expected value is taking your possible points, multiplying them by their probabilities, then adding the products.
You have a 0.001 chance of winning $5,000
You have a 0.003 chance of winning $2,000
You have a 0.996 chance of winning $0
EV=$5,000(0.001)+$1,000(0.003)+$0(0.996)=
$5 + $3 + $0 = $8
400
You have 5 skittles left, all different colors, in a bag of skittles. You are only going to eat three. This is the number of different ways can you eat them one at a time.
What is 60?
This is a permutation/combination problem because we are choosing from a group. It is a permutation because order matters.
n = the number you have to choose from
k=r= the number you are choosing
n = 5
k = 3
The equation for permutations is n!/(n-r)!
5!/(5-3)!
5!/2!
(5x4x3x2x1)/(2x1)
cancel out the numbers that are the same on top and bottom, and you are left with
5x4x3=60
400
This is the MAD of the following numbers:
5, 9, 11, 14, 16
What is 3.2?
First you have to find the mean:
(5+9+11+14+16)/5 = 55/5 = 11
Then you have to find your absolute deviations. This means you need to subtract every number from your mean, and then take the absolute value:
11-5 = 6
11-9 = 2
11-11 = 0
11-14 = -3 = 3
11-16 = -5 = 5
Then you need to find your SAD, which is the SUM of your absolute deviations
SAD = 6+2+0+3+5=16
Then you divide your SAD by the number of numbers you have to find your MAD
MAD = 16/5 = 3.2
500
There are 30 books in the class set of the book you are about to read in Literature. 7 of them are missing their cover. 6 others have been ripped. 3 others have been written in. You are 6th in line to pick a book, and all those who go before you pick books that are in good condition (no rips, writing, or missing covers). If you pick one randomly, this is the probability you get a good condition book.
What is 9/25?
You started with 30 books, with 16 in bad condition and 14 in good condition. The five people before you all take good condition books. So there are 25 books left total, and only 9 good condition books. So the probability is 9/25.
500
You have a bag of skittles with 10 red, 20 yellow, 15 green, 13 purple, and 7 orange skittles. You reach in the bag and grab a skittle. What is the probability you grab an orange one, decide you don't like orange and put it back, and then get a green skittle.
What is 1/6?
This is an independent problem. We know we are multiplying because of the use of the word AND THEN.
The probability of getting an orange skittle is 7/65
The probability of getting a green skittle is 15/65 = 3/13
7/65 x 3/13 = 21/845
500
A player tosses two coins into the air. He gains to win $1 times the number of heads that are obtained. However, he will lose $5 if neither coin is a head. This is the expected value of the game.
What is
Your expected value is taking your possible points, multiplying them by their probabilities, then adding the products.
Your chance of getting two heads is 1/4 (1/2*1/2), so this is your chance of getting $2
Your chance of getting one head is 1/2 (1/2*1/2 + 1/2*1/2), so this is your chance of getting $1
Your chance of getting no heads is 1/4 (1/2*1/2), so this is your chance of losing $5
You have a 1/4 chance of winning $2
You have a 1/2 chance of winning $1
You have a 1/4 chance of losing $5
EV=$2(1/4)+$1(1/2)+(-$5)(1/4)=
$1/2 + $1/2 + (-$5/4) = -$1/4
500
You have 5 close friends, but can only bring 3 to a concert with you. This is how many ways you can choose three friends.
What is 10?
This is a permutation/combination problem because we are choosing from a group. It is a combination because order doesn't matter (they just want to go)
n = the number you have to choose from
k=r= the number you are choosing
n = 5
k = 3
The equation for combinations is n!/[(n-r)!r!]
5!/[(5-3)!3!]
5!/(2!3!)
(5x4x3x2x1)/(2x1x3x2x1)
cancel out the numbers that are the same on top and bottom, and you are left with
5x4/(2x1)=10
500
This is the MAD of the following numbers:
6.96, 7.01, 7.11, 7.15, 7.17, 7.20
What is 0.077?
First you have to find the mean:
(6.96+7.01+7.11+7.15+7.17+7.20)/6 = 42.6/6 = 7.1
Then you have to find your absolute deviations. This means you need to subtract every number from your mean, and then take the absolute value:
7.1-6.96 = 0.14
7.1-7.01 = 0.09
7.1-7.11 = -0.01 = 0.01
7.1-7.15 = -0.05 = 0.05
7.1-7.17 = -0.07 = 0.07
7.1-7.20 = -0.1 = 0.1
Then you need to find your SAD, which is the SUM of your absolute deviations
SAD = 0.14+0.09+0.01+0.05+0.07+0.1 = 0.46
Then you divide your SAD by the number of numbers you have to find your MAD
MAD = 0.46/6 = 0.077