Theoretical and Experimental
Ali has 11 black shirts, 9 white shirts, 4 yellow shirts and 3 red shirts. If he selects a shirt from his closet at random, what is the probability he selects either a yellow shirt or a red shirt?
7/27
The probability that Ali selects either a yellow shirt or a red shirt is 7/27
Mr. Arruda has 3 shirts (red, blue, green), 2 pairs of pants (jeans, khakis), and 2 jackets (leather, denim). HOWEVER, he has a rule that the green shirt cannot be worn with khakis. Use a tree diagram to show all valid outfit combinations for Mr. Arruda. How many are there?
Image is provided.
There are 10 possible combinations
At a smoothie shop, customers can choose:1 fruit: banana, strawberry, mango AND 1 liquid: milk, juice.
Use an organized list to show all possible smoothie combinations (one fruit and one liquid).
How many different smoothies can be made?
There are 3 fruits × 2 liquids = 6 combinations
Banana + Milk
Banana + Juice
Strawberry + Milk
Strawberry + Juice
Mango + Milk
Mango + Juice
Describe an item that could be used to simulate each situation. Explain why each item is appropriate.
a) Choose one of two DVDs
b) Choose a sundae topping at random from 8 choices
c) Win a free Spotify subscription by collecting letters from pop bottle caps to spell C-A-P
ANSWERS CAN VARY
a) coin toss
A coin has two equally likely outcomes: heads and tails. You can assign Heads = DVD 1 and Tails = DVD This makes it a perfect and fair way to randomly choose between two options.
b) 8 section spinner
An 8-sided spinner has eight equal outcomes, just like the 8 topping choices. You could assign each number (1–8) to a topping.
c) A bag with equal numbers of letter tiles (like from Scrabble) containing C, A, and P
Drawing random letters from a bag mimics how you might find random letters under bottle caps. You could simulate the process of collecting letters over time to try to get all three required ones.
To introduce probability to her class, Ms. Segal wrote each letter of the word P-R-O-B-A-B-I-L-I-T-Y on a separate card and placed the cards face down.
a) What is the probability of choosing the letter B?
b) What is the probability of choosing a vowel? (A,E,I,O,U,Y)
c) What is the probability of choosing a consonant?
a) 2/11
b) 5/11 (yes Y)
c) 6/11 (not Y)
A retail store served 773 customers in October, and there were 44 complaints during that month.
Determine, as a percentage, the experimental probability that a customer submits a complaint.
Round your answer to the nearest whole percent.
44/773 --> 6%
A restaurant offers the following options:
Starter – soup or salad
Main – chicken, fish or vegetarian
Dessert – ice cream or cake
How many possible different combinations of starter, main and dessert are there?
Therefore there are 12 different combinations of starters, main and dessert.
(2x3x2) = 12
Using a Table, create an organized list of all the possible outcomes (sample space) when Cameron rolls two dice: one is 4-sided and the other is 6-sided
Image provided if needed.
Therefore, all the possible outcomes when Cameron rolls the two dice are as follows:
(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4), (4,1), (4,2), (4,3), (4,4), (5,1), (5,2), (5,3), (5,4), (6,1), (6,2), (6,3), (6,4)
Describe an item that could be used to simulate each situation. Explain why your item is appropriate.
a) How to determine which team starts the game: Team A or Team B.
b) A random student is selected from a class of 30 students.
c) A board game uses a 10-sided die to decide a player’s move (1–10). You don’t have that die — what could you use instead?
ANSWERS MAY VARY
a) a coin
b) a jar with 30 slips of paper
c) two 6-sided dice (use their sum up to 10, or re-roll)
4 marbles, numbered 1 to 4, are placed in a bag. Three marbles are removed one at a time. The three numbers are added. What sum is not possible?
A) 9
B) 5
C) 7
D) 6
B) 5
The sum that is not possible is the sum of 5
One person will be selected at random to represent the company at a special event. Below are the numbers of people who wish to be selected. From the data below:
Department Wearing Red Wearing Blue
A 0 5
B 6 1
C 2 10
D 12 4
a) What is the probability that a person from Department C will be selected?
b) What is the probability that someone wearing red will be selected?
a) The probability that a person from Department C will be selected is 12/40 --> 3/10 --> 30%
b) The probability that someone wearing red will be selected 20/40 --> 1/2 --> 50%
An ice cream parlor offers:
- 2 cone types: waffle, sugar
- 3 flavors: vanilla, chocolate, strawberry
- 3 toppings: sprinkles, syrup, gummy bears
Use a tree diagram to show all combinations of one cone, one flavor, and one topping. How many total combinations are there?
Image Provided.
Therefore there are 18 total combinations.
Isabel has four things to do everyday before school. For variety, she chooses slips of paper from a jar each morning to decide in which order she will do the jobs. What is the probability that Isabel will make the bed before walking the dog?
Things she has to do:
- Make the bed
- Walk the dog
- Eat breakfast
- Brush her teeth
12/24 --> 1/2
Therefore, Isabel has a 50% chance of making the bed before walking the dog.
Which model(s) could be used to determine the probability of each event: a coin, a bag of marbles, a deck of cards?
a) Melody selecting the only red jellybean in a bag of 26 jellybeans
b) Aaron scoring 10/10 on a true/false quiz
ANSWERS COULD VARY
a) bag of marbles
b) a coin
In his last 20 times at bat, Blake hit 12 home runs.
a) What is the probability of Blake hitting a home run at his next time at bat?
b) What is the probability of Blake not hitting a home run?
a) 12/20 or 3/5
b) 8/20 or 2/5
Two six-sided dice are rolled. The probability of rolling a sum of 5 is (1/9). What is the probability of not rolling a sum of 5?
A) 1/9
B) 7/9
C) 2/3
D) 8/9
D) 8/9
The probability of not rolling a sum of 5 is 8/9.
The probability of rolling a 5 is 4/36, which reduces to 1/9. 36-4 is 32/36 which reduces to 8/9
Artin and Farsam are snowboarding in the Rockies. On one run down the mountain, they decide to flip a coin to choose which of two paths they will take at each of the three places where the ski runs branch. They will go down the left ski run if the coin is a head and the right ski run if the coin is a tail.
a) What is the probability that they will take Thunder Road?
b) What is the probability that Artin and Farsam will finish on a run containing the name Bowl?
c) What is the probability that they will take Thunder Road and Quick Break? Explain your answer
a) 50% (4/8 -->1/2)
b) 2/8 --> 1/4 --> 25%
c) because only one path satisfies the answer of including Thunder Road AND Quick Break, the answer is 1/8 --> 12.5%
Naomi rolls two dice. Each die has numbers 1–6. List all outcomes Naomi has where the sum is greater than 9 and the first die shows an even number.
*Make sure there are no repeating numbers! For example, (5,3) is the same as (3,5) so it counts as one outcome, not two (in this case).
Image Provided
4 total (non-repeating) outcomes
Therefore, all outcomes Naomi has where the sum is greater than 9 and the first die shows an even number is 4/36 --> 1/9 --> 11.1%
Which model is not appropriate for this probability simulation?
Abigail is driving through a small village. At each intersection, she has three choices of direction: left, right, and straight. Only one of these choices leads to where she wants to go. What is the probability that Abigail will choose the correct direction at the next two intersections?
A) 10-sided dice
B) 6-faced dice
C) a coin
D) a 3-sided spinner
C) a coin
The probability that Abigail will choose the correct direction at the next two intersections is 1/2 because there are two intersections left, and therefore only two options remain. Therefore the appropriate model for this probability is the coin.
Jiyu flips a coin and rolls a number cube (die) at the same time. She repeats this experiment 60 times and records that:
She gets Heads and an even number 22 times.
a) What is the theoretical probability of getting heads and an even number?
b) What is the experimental probability based on Jiyu’s data?
Theoretical Probability = ¼ or 0.25
Experimental Probability ≈ 11/30 or 0.367
There are 18 girls and 12 boys in a class. 2/9 of the girls and 1/4 of the boys walk to school. One of the students who walks to school is chosen at random to receive a "walk to school prize". Find the probability that the student is a boy.
Answer: 3/7
2/9 of 18=4 (number of girls)
1/4 of 12=3 (number of boys)
4 +3=7 (total number of students who walk to school)
7 students walk to school. 4 are girls and 3 are boys. Therefore, the probability the student is a boy is 3/7
Cyrus and Borna cannot decide which sport they want to play at recess.
Cyrus suggests that they roll a number cube and spin a spinner. The spinner is divided in half. One half is labelled 1 and the other is labelled 2. If the sum of the numbers from the cube and spinner is even, Cyrus gets to decide. Otherwise, Borna does.
Borna suggests they toss a coin and roll a colour cube. Three faces of the cube are yellow and three faces are blue.
a) Draw a Tree Diagram to show all the possible outcomes for each person's suggestion for deciding
b) What is the theoretical probability of Borna winning using Cyrus's suggestion?
Answer is an image
b) 1/2 --> 50%
Luna and Chanel create a Bingo Pingo game for their probability class. There are 25 cards of each color (red, blue, green, and yellow). Each color has 19 number cards, consisting of: 1 zero AND Two of each number from 1 to 9.
Use an organized list to show all the number cards in the red color set. How many cards are there total?
0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9
Therefore, there are 19 cards.
For each scenario below, suggest a simple item (or method) that could be used to simulate the situation. Clearly explain how it models the probabilities involved.
a) A game show contestant opens a prize door. There are 3 doors: one has a car, the others have goats.
b) You randomly win a prize 1 out of every 12 candy bars. Simulate this situation.
c) In a game, players roll two dice. If the sum is greater than 9, they win. Describe how to simulate this outcome without using actual dice.
ANSWERS MAY VARY
a) 3-sided spinner or number cards 1–3
Each door has a 1 in 3 chance of having the car. Assign 1 = car, 2 & 3 = goats. Randomly spin or draw to simulate choosing a door.
b) 12-sided number spinner or number cards 1–12
One number (e.g., 7) to be the winning number. The rest are non-winners. Spin or draw to see if you win.
c) Use a random number generator (or app) to generate two numbers between 1 and 6.
Simulate two dice rolls using digital tools or number cards 1–6, sum them, and check if the total is greater than 9. This models the same chance without needing physical dice
In a game, Lucas draws one marble from a bag containing 3 red, 2 blue, and 1 green marble. Without replacing it, he draws a second marble.
a) Draw a tree diagram to show all the possible outcomes.
b) What is the probability that both marbles Lucas has drawn are red?
a)
R → R
R → B
R → G
B → R
B → B
B → G
G → R
G → B
(Total of 6 × 5 = 30 unique ordered outcomes; only 8 branches are needed if grouped by first pick and second.)
b) Therefore the probability that both marbles Lucas has drawn are red is 1/5 or 0.2