Probability
There is a large branch of mathematics that studies the theoretical probability of events. As basic examples, coins, throwing dice, a deck of cards, spinners, each have a discrete number of possible outcomes. We can study the theoretical probabilities before running experiments to verify the predicted final results.
In fields such as biology, knowing the genetic alleles of a plant or vegetable correctly can then result on proper proportions of desired crops. A farmer can choose seeds that create 25% yellow healthy peapods, and 70% green ones based on the theoretical compositions of the genetics within the seeds.
In game theory, simulations of games or events of choice can be mapped also with “sample spaces”
which are a map of all possible events from the given materials in the experiment.
Coins
Possible Outcomes from a coin toss: {H,T}
Sample Space 24= 16 events 4 Heads, 0 Tails H H H H
3 Heads, 1 Tail H H H T
H H T H
H T H H
T H H H
2 Heads, 2 Tails H H T T
H T H T
H T T H
T H H T
T H T H
T T H H
1 Head, 3 Tails T T T H
T T H T
T H T T
H T T T
0 Heads, 4 Tails T T T T
The table on the left is a map of all possible outcomes from tossing 4 coins in a probability experiment.
This is useful to see that for each toss the outcome has two, and only two, possible results. We call these type of events BINOMIAL PROBABILITY events.
For each “fair” coin the probability of:
“Heads” = H = 1/2 = 50%
“Tails” = = T 1/2 = 50%
The distribution shows that it is much rarer to have all 4 coins the same than different. The total PDF:
Heads 4 3 2 1 0
Tails 0 1 2 3 4
Probability 1
16
164
166
164
161
The 2009 Lincoln Penny featured 4 collectable designs on reverse side.
Four coins are tossed.
What are the following probabilities ?
Q1: P(4 tails) ? 1 event
16 possible events
=
161
Q2: P(3 heads) ? 4 events
16 possible events
=
41Q3: P(2 or more heads) ? (1 + 4 + 6) events
16 possible events
=
1611The sample space is a list of
all possible events.
The list on the left has all 16 possible tosses
with 4 coins.
Spinners are usually circle shaped, 2 dimensional tools for random events. With a flick of the 4 finger, an arrow spins and lands in a labelled section of the circle. Also, The spaces themselves can be unique or repeat for various probabilities. Also, they can be equally spaced, or have sectors with different angle measures.
A spinner pictured at right, on top, has 8 equal sections. What are the following probabilities ?
Q1: P(Number 2) ? 4 events
8 possible events
=
21
Q2: P(Odd Number) ? 4 events
8 possible events
=
21Q3: P(3 or less) ? 3 events
8 possible events
=
83Spinners
A spinner pictured at right, on bottom, has 8 equal sections. What are the following probabilities ?
Q1: P(Number 2) ? 3 events
8 possible events
=
83Q2: P(Odd Number) ? 2+3 events
8 possible events
=
85Q3: P(3 or more) ? 3 events
8 possible events
=
83
Throwing dice are commonly used to simulate random events. The nice thing about 6 sided dice is that they can be used for 2,3 or 6 different outcomes by grouping the results onto pairs or triplets.
Also, the math of throwing two or more dice and adding the sums simulates new distributions of compound events.
Dice
Sample Space:
All possible Events adding 2 Dice
Sum 1 2 3 4 5 6
1 2 3 4 5 6 7
2 3 4 5 6 7 8
3 4 5 6 7 8 9
4 5 6 7 8 9 10
5 6 7 8 9 10 11
6 7 8 9 10 11 12
A pair of standard dice are tossed. What are the following probabilities ? Q1: P(Seven) ?
6 event
36 possible events
=
61
Q2: P(Even Sum ) ? 18 events
36 possible events
=
12Q3: P(Sum 9 or more) ? 10 events
36 possible events
=
185
Standard Deck of Cards
13 Values, * 4 Suits = 52 Total Cards
♣ Clubs A 2 3 4 5 6 7 8 9 10 J Q K
♦ Diamonds A 2 3 4 5 6 7 8 9 10 J Q K
♥ Hearts A 2 3 4 5 6 7 8 9 10 J Q K
♠ Spades A 2 3 4 5 6 7 8 9 10 J Q K
Picture of a popular brand of cards:
Pictures of a few cards {Ace of Spades, Joker, Jack of Diamonds, Queen of Hearts, King of Clubs}
The “joker” is the 53rd card in a standard deck, but it is not included in card play in most games.
It was added to as a Trump card in the Civil War for the game of Euchre.
A card is drawn from a deck of cards. What are the following probabilities ?
Q1: P(♦ Diamond) ? 13 event
52 possible events
=
14
Q2: P(Face card) ? 12 events
52 possible events
=
133Q3: P(Face & ♦ Diamond) ? 3 events
52 possible events
=
523
SHSAT Lesson #26 Classwork: Probability
1) Two coins are flipped. What is the probability that they are both tails ?
A) 0 B) 0.25 C) 0.50 D) 0.75 E) 1 2) What is the probability of rolling a 5 sum with a pair of dice
{1,2,3,4,5,6} each?
A) 1/12 B) 1/10 C) 1/9 D) 1/6 E) 1/4 3) A spinner has 8 equal spaces numbered {1,2,3,4,5,6,7,8}. What is the
probability of the spinner landing on a prime number ?
A) 0 B) 1/8 C) 1/4 D) 3/8 E) 1/2 4) What is the probability of drawing a “three” from a standard set of
52 playing cards ?
Standard deck with 4 suits each with 1 to 10, Jack, Queen, King)
(
(See page three for the full card deck)
A) 1/4 B) 1/5 C) 1/6 D) 1/13 E) 1/52
5) The 12 face cards are separated from a deck of cards. Hearts & Diamonds {♥,♦} are RED cards. Clubs & Spades {♣,♠} are BLACK cards.
{ J♣, J♦, J♥, J♠, Q♣, Q♦, Q♥, Q♠, K♣, K♦, K♥, K♠}
What is the probability that a black queen is picked from this subset ?
A) 1/12 B) 1/6 C) 1/4 D) 1/3 E) 1/2
6) A standard die is rolled three times. What is the probability of at least 2 sixes ? A) 1/216 B) 1/36 C) 1/6 D) 2/27 E) 5/72
7) A principal puts the following names into a hat {Harriet, George, Franklin, Edward, Donna, Charles, Betty and Anne}. What is the probability that a female with a name that ends in a vowel is chosen ?
A) 1/8 B) 1/4 C) 3/8 D) 1/2 E) 3/4 8) The probability of rain in Utopia is 30% everyday. What is the probability that
it rains on Saturday but NOT on Sunday ?
A) 10%
B) 21%
C) 25%
D) 40%
E) 50%
9) A jar has 6 black and 4 white marbles. If two marbles are drawn
from the jar, what is the probability that they are BOTH black?
A)
2410*
249B)
106*
96C)
106*
95D)
106*
105E)
104*
9310) What is the probability of rolling a 7 or 11 with a pair of dice ? A) 2/11 B) 1/2 C) 1/5 D) 2/9 E) 1/3
11) If the probability that is will rain on Saturday is 30% and the
probability that it will not rain on Sunday is 60%, what is the probability that it will rain on Saturday and NOT rain on Sunday ?
A) 10%
B) 12%
C) 18%
D) 30%
E) 70%
12) What is the probability of drawing a card that is at least 3 but at the most 7 from a deck of 52 playing cards ?
A) 3/13 B) 4/13 C) 5/13 D) 6/13 E) 10/13
13)A card is drawn from a standard deck of playing cards.
What is the probability that the card is both a face card {Jack, Queen or King} and a diamond (♦ )suit ?
A) 1/52 B) 1/26 C) 3/52 D) 1/13 E) 5/52
14) A baseball pitcher has a 80% chance of striking out a batter. What is the probability that the pitcher has 3 strikeouts in a row ?
A) 64%
B) 80%
C) 24%
D) 8%
E) 51.2%
Classwork Problem Set: (CONTINUED)
15) A basketball player has a 24% chance of making a 3 point shot. The same player has a 50% chance of making a 2 point shot. He also has a 75%
chance of making a foul shot, worth 1 point each. Which statement is TRUE ?
A) He would probably make more points with ten 3 point shots than ten 2 point shots.
B) He would probably make more points with fifteen 3 point shots than fifteen 2 point shots.
C) He would probably make more points with fifty 3 point shots than forty 2 point shots.
D) He would probably make more points with fifty 3 point shots than thirty 2 point shots.
E) He would probably make more points with fifty 3 point shots than sixty 2 point shots.
16) What is the least number of handshakes needed if 7 people are in a room, so that each person has met everyone else ?
A) 49 B) 15 C) 21 D) 28 E) 42 17)There are 10 closed lockers in a hallway. There are 10 people, each
given a tee-shirt numbered 1 to 10. An exercise with instructions follows:
“Go only to the lockers the are a multiple of your shirt number.
If it is open, close it.
If it is closed, open it.”
Person #1 goes first, and opens every locker.
Person #2 goes next and closes every 2nd locker.
Person #3 goes next and opens/closes every 3rd locker.
Person #4 goes next and opens/closes every 4th locker.
… and so on.
After all 10 people have gone, how many lockers are open?
A) 0 B) 3 C) 6 D) 7 E) 10
18) Three dice are thrown. What is the probability that the sum is less than 5 or greater than 16 ?
A) 1/216 B) 2/216 C) 4/216 D) 8/216 E) 16/216
Classwork Problem Set: (CONTINUED)
19) A meal consists of one choice of each of the following:
chicken noodle, split pea, alphabet soup or salad}
{ {corn, string beans or rice}
fish, BBQ chicken or Steak}
{{cake, ice cream or tiramisu}
How many meals are possible ?
A) 13 B) 81 C) 108 D) 162 E) 192
20) A dance is created with a single step in chosen from one of four directions, then repeated 2 more times, with the same 4 choices. How many dance steps are possible ?
A) 12 B) 24 C) 128 D) 64 E) 120
21) Meg dresses herself from her dresser with 5 different pair of socks, 3 different jeans, and 6 different blouses. How many unique outfits does she have ?
A) 24 B) 48 C) 60 D) 90 E) 150
22) If a coin is flipped twice, and a 5 section spinner with sections {Red, Orange, Yellow, Green and Blue} is then spun. What is the probability that one head & one tail is flipped, and then the spinner does NOT land on Red or Green ?
A) 30%
B) 45%
C) 50%
D) 60%
E) 85%
23) A six sided die and an 8 section spinner numbered 1 to 8 are played. What is the probability that the sum is double digits ?
A) 15/48 B) 12/48 C) 9/48 D) 6/48 E) 3/48
24) A spinner has twelve sections labelled as follows:
{R, Y, B, R, Y, B, R, Y, G, R, Y, B}
What is the probability that it lands on R or G ?
A) 4/12 B) 5/12 C) 7/12 D) 8/12 E) Impossible.
Classwork Problem Set: (CONTINUED)
Washington Vs. Jefferson Middle School Washington
Likes Regents
Doesn’t Like
Regents Total
Regents 73 15 88
No Regents 10 2 12
Total 83 17 100
Jefferson
Likes Regents
Doesn’t Like
Regents Total
Regents 158 19 177
No Regents 18 5 23
Total 176 24 200
25) How many more regents students does Jefferson Middle school have? A) 83 B) 85 C) 87 D) 89 E) 91 26) Given that a student is taking the regents at Washington, what
proportion voted that they like their Regents class? A) 83/100 B) 73/88 C) 10/12 D) 15/88 E) 73/83 27) What proportion of the students in BOTH middle schools who take Regents class do
not like the class? A) 34/255
B) 41/300 C) 41/265 D) 34/111 E) 34/265 28) At Washington Middle School, what is the probability that a student who is not
taking regents class does not like regents class ? A) 12%
B) 14%
C) 16.67%
D) 18%
E) 24%
29) If you pass the regents you get credit for 9th grade in that subject area. If the probability is 80% that a regents student at Jefferson (177 total) will pass the regents, approximately how many students will skip a 9th grade class the following year ?
A) 90 B) 12 C) 142 D) 155 E) 165 30) The probability of a Washington student passing the Math Regents is 85%. Also, it’s
estimated at 95% for the Living Environment Regents. If three-quarters of the Washington Regents Students take Math Regents and the other one-quarter take Living Environment, then predict approximately how many students will pass.
A) 60 B) 65 C) 77 D) 84 E) 93
SHSAT Lesson #26 Homework: Probability
1) Two coins are flipped. What is the probability that they is at least one tail ?
A) 0 B) 0.25 C) 0.50 D) 0.75 E) 1 2) What is the probability of rolling a sum of 9 with a pair of dice ? A) 1/12
B) 1/10 C) 1/9 D) 1/6 E) 1/4 3) A spinner has 8 equal spaces numbered {0,1,2,3,4,5,6,7}. What is the
probability of the spinner landing on a number less than six?
A) 7/8 B) 5/6 C) 3/4 D) 2/3 E) 1/2 4) What is the probability of drawing the “jack of clubs (J♣)” from the
black cards only. Clubs (♣) and Spades (♠) are the black playing cards ?
A) 1/4 B) 1/5 C) 1/13 D) 1/26 E) 1/52 5) The 12 face cards are separated from a deck of cards.
{ J♣, J♦, J♥, J♠, Q♣, Q♦, Q♥, Q♠, K♣, K♦, K♥, K♠}
What is the probability that a king is picked from this subset ?
A) 1/12 B) 1/6 C) 1/4 D) 1/2 E) 1/3
6) A die is rolled three times. What is the probability of at least 2 threes ?
A) 1/216 B) 1/36 C) 1/6 D) 2/27 E) 5/72
7) A principal puts the following names into a hat {Harriet, George, Franklin, Edward, Donna, Charles, Betty and Anne}. What is the probability that a person with a name that ends in a vowel or “y” is chosen ?
A) 1/2 B) 1/4 C) 3/8 D) 1/8 E) 3/4
8) The probability of rain in Utopia is 40% everyday. What is the
probability that it rain 3 consecutive days in a row ?
A) 3/250 B) 8/125 C) 33/500 D) 16/125 E) 4/125
Homework Problem Set: (CONTINUED)
9) A jar has 7 red, 6 white and 5 blue marbles. If two blue, three red and four white marbles are removed, what is the probability of drawing a red marble from the remaining set ?
A) 4/7 B) 3/9 C) 2/14 D) 5/11 E) 4/9 10) What is the probability of rolling a number greater than a sum of 7 with a
pair of dice ? A) 1/12
B) 2/12 C) 3/12 D) 4/12 E) 5/12
11) If there is a 20% of snow on Thursday, but 70% chance of snow on Friday, what is the probability that it will NOT snow on Thursday but that it WILL snow on Friday ?
A) 56%
B) 65%
C) 75%
D) 90%
E) 14%
12) What is the probability of drawing a card that is NOT a face card from a deck
of playing cards ? A) 10/13
B) 12/13 C) 8/13 D) 3/13 E) 1/13
13) Two cards are drawn from a deck of cards, with replacement.
What is the probability that at least one card is a diamond (♦) suit ?
A) 2( )( )5213 5239 + (52 523)( )11 B) 2( )( )5213 5239 + (5213)( )5113 C) 2( )( )5213 5239 + (5213)( )5212 D) 2( )( )5213 5139 + (5213)( )5112 E) 2( )( )5213 5239 + (5213)( )5213
14) A baseball pitcher has a 85% chance of striking out a batter. What is the
probability that the pitcher has 4 strikeouts in a row ? A) 85%
B) 66.67%
C) 52.2%
D) 47.8%
E) 40%
Homework Problem Set: (CONTINUED)
15) A basketball player has a 30% chance of making a 3 point shot. The same player has a 65% chance of making a 2 point shot. Which statement is TRUE ?
A) He would probably make more points with ten 3 point shots than ten 2 point shots.
B) He would probably make more points with fifteen 3 point shots than fifteen 2 point shots.
C) He would probably make more points with fifty 3 point shots than forty 2 point shots.
D) He would probably make more points with fifty 3 point shots than thirty 2 point shots.
E) He would probably make more points with fifty 3 point shots than sixty 2 point shots.
16) What is the least number of handshakes needed if 8 people are in a room,
so that each person has met everyone else ? A) 7
B) 15 C) 21 D) 28 E) 36 17) There are 27 closed lockers in a hallway. There are 27 people,
each given a tee-shirt numbered 1 to 27. An exercise with instructions follows:
“Go only to the lockers the are a multiple of your shirt number.
If it is open, close it.
If it is closed, open it.”
Person #1 goes first, and opens every locker.
Person #2 goes next and closes every 2nd locker.
Person #3 goes next and opens/closes every 3rd locker.
Person #4 goes next and opens/closes every 4th locker.
… and so on.
After all 27 people have gone, how many lockers are open?
A) 3 B) 5 C) 7 D) 11 E) 15
18) Three dice are thrown. What is the probability that the sum is less than 6 or
greater than 15 ? A) 10/216
B) 20/216 C) 40/216 D) 80/216 E) 160/216
Homework Problem Set: (CONTINUED)
19) A meal consists of one choice of each of the following:
chicken noodle, split pea, alphabet soup or salad}
{
corn, string beans or rice}
{
fish, BBQ chicken or Beefsteak}
{
cake, ice cream or tiramisu}
{
How many meals with Beefsteak are possible ?
A) 108 B) 54 C) 48 D) 36 E) 27
20) A dance is created with a single step chosen from one of four directions, then repeated 3 more times, with the same 4 choices for each move. How many dance steps are possible ?
A) 256 B) 128 C) 64 D) 32 E) 16
21) Meg dresses herself from her dresser with 8 different pair of socks, 5 different jeans, and 9 different blouses. How many unique outfits does she have ?
A) 360 B) 270 C) 180 D) 90 E) 46
22) If a coin is flipped 10 times, what is the probability that it will
be face up 10 heads in a row ? A) 1/1,024
B) 1/100 C) 1/256 D) 1/512 E) 1/64
23) A six sided die and an 8 section spinner numbered 1 to 8 are
played. What is the probability that the sum is single digits ? A) 45/48 B) 36/48 C) 39/48 D) 42/48 E) 33/48
24) A spinner has twelve sections labelled as follows:
{R, Y, B, R, Y, B, R, Y, G, R, Y, B}
What is the probability that it does NOT land on B or G ?
A) 4/12 B) 5/12 C) 7/12 D) 8/12 E) Impossible.
Jefferson Vs. Lincoln High School Jefferson
Likes AP Class
Doesn’t Like
AP Class Total
AP Class 63 15 78
No AP Classes 20 2 22
Total 83 17 100
Lincoln
Likes AP Class
Doesn’t Like
AP Class Total
AP Class 144 24 168
No AP Classes 18 14 32
Total 162 38 200
25) How many more Advanced Placement (AP) students does Lincoln High
school have? A) 60
B) 70 C) 80 D) 90 E) 100 26) Given that a student is taking the AP Class at Jefferson, what proportion
voted that they like their class? A) 78/100
B) 83/100 C) 20/22 D) 15/78 E) 63/78 27) What proportion of the students in BOTH schools who take AP class do
not like the class? A) 39/246
B) 63/246 C) 207/246 D) 39/300 E) 146/300 28) At Jefferson, what is the probability that a student who is not taking AP
class does not like AP class ? A) 8.3%
B) 9.1%
C) 10%
D) 12%
E) 22%
29) If you pass the AP you get credit a year of College Credit in that subject area. If the probability is 80% that an AP student at Jefferson will pass the AP, approximately how many AP students will earn a year of College Credit class ?
A) 78 B) 62 C) 48 D) 43 E) 35 30) The probability of a Washington student passing the AP Math 1 is 85%.
Also, it’s estimated at 95% for AP US History. If three-quarters of the Washington Students take AP Math 1 and the other one-quarter take AP US History, then predict approximately how many students will pass.
A) 125 B) 133 C) 147 D) 168 E) 155
Extra Credit Challenge Questions:
1) A rectangular floor that is 10 feet wide and 17 feet long is tiled with 170 one foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit ?
A) 17 B) 25 C) 26 D) 27 E) 28 2) For some positive integer k, the repeating base k representation of
the (base ten) fraction is 517 0.23 k= 0.2323232323... . k What is k ? A) 13 B) 14 C) 15 D) 16 E) 17 3) A child builds towers using identically shaped cubes of different
color. How many different towers with a height 8 cubes cn the child build with 2 red cubes, 3 blue cubes, and 4 green cubes ? (one cube will be left out.)
A) 24 B) 288 C) 312 D) 1,260 E) 40,320 4) What is the least possible value of
x )(x )(x )(x ) 019 ( + 1 + 2 + 3 + 4 = 2
When x is a real number ?
A) 2017 B) 2018 C) 2019 D) 2020 E) 2021
5) Alicia had two containers. The first was five-sixths full of water and the second was empty. She poured all the water from the first container into the second container, at which point the second container was three quarters full of water. What is the ratio of the volume of the first container to the second container ?
A) 5/8 B) 4/5 C) 7/8 D) 9/10 E) 11/12
AMC 10 2019 contest. FORM A {10,18,17,19} FORM B {1}