Probability of Dependent and Independent Events

Multiplication

Rule of

Probability

Let's start by reviewing some concepts related to probability models.

Match

A Double Venn Diagram shows information related to sets.

Match each diagram with its correct set notation.

• $$A$$

• $$B$$

• $$A\cap B$$

• $$B'$$

• $$A\cup B$$

Hotspot

The Venn Diagram shows the universal set with 50 responses.

Indicate where the diagram shows $$B\cap C.$$

Math Response

The Venn Diagram shows the universal set with 50 responses.

Determine $$P\left(B\ and\ C\right).$$

Write your answer as a fraction.

Type answer here

Correct Answer: \frac{2}{50}

Deg^°

Rad

Hotspot

The Venn Diagram shows the universal set with 50 responses.

Indicate the location(s) on the Diagram that shows: the complement of event B, or $$\overline{B}.$$

Math Response

The Venn Diagram shows the universal set with 50 responses.

Determine $$P\left(B'\right),$$ the probability of the complement of B.

Write your answer as a fraction.

Type answer here

Correct Answer: \frac{24}{50}

Deg^°

Rad

Labelling

A guidance counselor is planning schedules for 30 students. 

16 want to take Spanish and 11 want to take Latin.  Five students want to take both. 

Display this information on the Venn diagram by labeling the correct value for each location in the Venn Diagram.

Drag labels to their correct position on the image

11

16

6

5

30

8

Multiple Choice

Amanda spins a spinner numbered 1 to 10.

What is the probability that the spinner lands on an odd number or a number that is divisible by 3?

• $$\frac{4}{5}$$

• $$\frac{3}{5}$$

• $$\frac{1}{5}$$

• $$\frac{3}{20}$$

​

Dropdown

Two events are ​

Select answer

when the probability of one event happening does not affect the probability of the other event.

We can represent the
concept of flipping a coin twice with a tree diagram like the one shown.

We multiply the probabilities along the branches to find the overall probability of one event AND the next event occurring.

Multiple Choice

When flipping two fair coins, what is the probability of getting tails both times?

• $$1$$

• $$\frac{1}{2}$$

• $$\frac{1}{4}$$

• $$\frac{1}{8}$$

The probability of getting two "tails" in a row would be:

When two events are independent, we can say that

One way of proving if events are independent or dependent is by checking if P(A and B) = P(A)*P(B).

Multiple Choice

Consider rollling a pair of six sided number cubes with the numbers 1,2,3,4,5,6 on the faces.

Find the probability of both cubes landing on a $$3.$$

• $$\frac{1}{3}$$

• $$\frac{1}{6}$$

• $$\frac{1}{12}$$

• $$\frac{1}{36}$$

Multiple Choice

In a fair deck of 52 cards, half of the deck are red cards.

The other half of the deck are black cards.

Which expression represents the probability of pulling two black cards without replacement?

• $$\frac{26}{52}+\frac{26}{52}$$

• $$\frac{26}{52}+\frac{25}{51}$$

• $$\frac{26}{52}\cdot\frac{25}{51}$$

• $$\frac{26}{52}\cdot\frac{26}{52}$$

Categorize

Options (4)

Organize these options into the correct categories.

Independent Events

Dependent Events

Multiple Choice

A table of 5 students has 3 seniors and 2 juniors.

The teacher is going to pick 2 students at random from this group to present homework solutions.

Which expression can be used to find the probability that both students are juniors?

• $$\frac{2}{5}\cdot\frac{2}{5}$$

• $$\frac{2}{5}\cdot\frac{1}{5}$$

• $$\frac{2}{5}\cdot\frac{1}{4}$$

• $$\frac{2}{5}+\frac{1}{4}$$

Multiple Choice

A key ring has 12 keys, 3 of which open the house door.

Two keys are randomly selected without replacement.

What is the probability that neither key opens the door?

• $$\frac{1}{22}$$

• $$\frac{6}{11}$$

• $$\frac{9}{16}$$

• $$\frac{21}{22}$$

Variation of the General Multiplication Rule

Given any two unknowns, you can solve for the third unknown using this rule.

Multiple Choice

Given: $$P(A)=\frac{4}{9}$$ and $$P(A\ and\ B)=\frac{2}{9}$$ find $$P\left(B\mid A\right)$$

Hint: $$P\left(B\mid A\right)=\frac{P\left(A\ and\ B\right)}{P\left(A\right)}$$

• $$\frac{1}{2}$$

• $$\frac{1}{9}$$

• $$\frac{1}{3}$$

• $$\frac{2}{9}$$