Probability

​PROBABILITY

Basic Probability

Learning Objective: Students will determine the probability of an event.

Language Objective: Students will express their reasoning in written form.

Vocabulary

Probability: the likelihood of something happening. It is expressed as a number between 0 (will never happen) and 1 (will always happen). It can be expressed as a decimal, fraction, percentage or ratio.

Vocabulary

Experiment: a situation involving chance that leads to the result called the outcome.

Outcomes: the result of a single trial of an experiment.

Event: one or more outcomes of an experiment.

Sample Space: the set of all possible outcomes of that experiment.

Examples

Example 1: What is the sample space for choosing a prime number less than 15 at random?

Example 2: What is the sample space for choosing 1 jelly bean at random from a jar containing 5 red, 7 blue, and 2 green jelly beans?

Example 3: What is the sample space for choosing 1 letter at random from the word DIVIDE?

Multiple Choice

A bag contains two red marbles, three blue marbles, and four green

marbles. What is the sample space of the marble bag?

• red, blue, green

• 2, 3, 4

• 2 red, 3 blue, 4 green

• red, red, blue, blue, blue, green, green, green, green

Fundamental Counting Principle

States that if event x can occur in x ways and event y can occur in y ways, then x followed by y can occur in x×y ways. Or in other words, to find the total outcomes possible you take the total outcome from the first event and you multiply it by the total outcome of the second event.

Example 4

The cafeteria has 4 types of sandwiches, 6 types of chips and two types of drinks. How many possible lunch combos are there?

Multiple Choice

Linda is trying to pick out an outfit for school. She has 4 tops, 3 pants, and 5 shoes to pick from. How many total possible outfit combinations does she have?

• 3

• 5

• 12

• 60

Calculating Probability

Calculating Probability

Formula:

Example 5

A bag contains 4 blue marbles, 6 green marbles and 3 yellow marbles. A marble is drawn at random from the bag.

a) What is the probability of drawing a green marble?

b) What is the probability of drawing a yellow marble?

c) What is the probability of drawing a blue marble?

d) What is the probability of drawing a red marble?

Multiple Choice

What is the probability of rolling an even number on a die?

• 0.25

• 0.5

• 0

• 1

Multiple Choice

In his Halloween bag, Jack got 3 Twix, 8 Snickers, 5 Lollipops, and 4 Reese’s. Jack pulls candy out of his bag at random. What is the probability of Jack pulling out a Twix or a Snickers on one try?

• $$\frac{11}{20}$$

• $$\frac{24}{20}$$

• $$\frac{11}{40}$$

• $$\frac{24}{400}$$

Fill in the Blanks

Ahmed spins the spinner shown below once. Each sector shown has an equal area.

What is P(shaded area)?

Tree Diagram and Probability

Tree Diagram

A tree diagram is a branching structure that maps the outcomes of chances/probabilities.

The more branches, the more outcomes.

Example:

Take flipping a coin. Your only outcomes are heads and tails. So your tree would look like:

Flipping a Coin Twice:

If you flip the coin a second time you still only have the outcomes heads and tails but this time they branch out from you previous outcome. So your tree diagram looks like this:

Notice: The Sample Space. {HH, HT, TH, TT}

Example: Sample Space

You are trying to decide what to eat for lunch. The cafeteria offers a ham, turkey or peanut butter sandwich and cool ranch or nacho

cheese Doritos. What is the sample space?

Multiple Choice

Which shows the sample space for flipping two coins?

• H, T

• HH, HT, TH, TT

• HH, TT

• Dime, Nickel

Multiple Choice

What is the sample space when you roll a die and then flip a coin?

• 1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T

• 2

• 1,2,3,4,5,6,H,T

• 12

Multiple Choice

The cafeteria is serving three kinds of sandwiches: tuna (T), chicken, (C), and peanut butter (P). They are also serving a choice of two drinks: milk (M) or water (W). Which shows the sample space of possible combinations?

• TW, CW, PW

• TW, TM, TC, CW, PW

• TW, TM, CW, CM, PW, PM

• TCP, MW

Find Probabilities:

To calculate the overall probabilities:

- We multiply probabilities along branches.

- We add probabilities down columns.

We can now see things like:

• The probability of getting heads then heads is 25%.

• All the probabilities add up to 1 or 100% .

• The probability of getting AT LEAST one head from two tosses is 0.25 + 0.25+ 0.25 = 0.75

Example: Tree Diagram

You go to a restaurant where you are able to create your own Pizza. Use the following to create your own pizza options: Crust {Thin, Stuffed}, Size {Medium. Large, X-Large}, 1-Topping {Pepperoni, Cheese, Buffalo chicken}

Create a tree diagram illustrating all the possible pizzas that can be made.

1. P(thin, medium, cheese)

2. P(stuffed, large, pepperoni)

3. P(thin, large, pepperoni or cheese)

4. P(stuffed, X-large, cheese or buffalo chicken)

Multiple Choice

What is the probability of randomly choosing a breakfast with a bagel and coffee?

• 1/8 because there is only 1 outcome out of the 8 possible choices that include both a bagel and coffee.

• 1/2 because there is only cereal and a bagel to choose from

• 1/4 because there is only one out of the four choices that include coffee

• 3 because the words "coffee" and "bagel" are on the diagram a total of 3 times.

Multiple Choice

Use the tree chart to find the probability of randomly picking an outfit with a white shirt and jeans.

• There are 6 outfits

• 2/6, or 2 out of the 6 outfits have a white shirt with jeans

• 3/6, or 3 out of the 6 outfits have a white shirt with jeans

• 1/6, or 1 out of the 6 outfits has a white shirt with jeans.

Multiple Choice

The tree diagram shows the outcomes of rolling a die and flipping a coin.

What is the probability of rolling an even number and flipping a head?

• ³/₁₂

• ¹/₂

• ⁴/₁₂

• ⁰/₁₂