# Conditional Probability

23 questions

# Conditional Probability

##### She Made Some Tarts

Noah will select 1 card at random from a standard deck of cards. Find the probabilities. Write your answers as fractions in lowest terms.

1) $P \text{(the card is a queen)}$

2) Explain or show your reasoning.

3) $P \text{(the card is a heart)}$

4) Explain or show your reasoning.

5) $P\text{(the card is a queen and heart)}$

6) Explain or show your reasoning.

7) Elena pulls out only the hearts from the deck and sets the rest of the cards aside. She shuffles the hearts and draws one card. What is the probability she gets a queen?

##### Under One Condition

Kiran notices that the probabilities from the warm-up can be arranged into at least two equations.

$P \text{(the card is a queen and heart)} = P\text{(the card is a queen | the card is a heart)} \cdot P\text{(the card is a heart)}$ since $\frac{1}{52} = \frac{1}{13} \cdot \frac{13}{52}$.

$P \text{(the card is a queen and heart)} = P\text{(the card is a heart | the card is a queen)} \cdot P\text{(the card is a queen)}$ since $\frac{1}{52} = \frac{1}{4} \cdot \frac{4}{52}$.

Kiran wonders if it is always true that $P\text{(A and B)} = P\text{(A | B)} \cdot P\text{(B)}$ for events A and B. He wants to check additional examples from drawing a card from a deck.

8) If Event A is “the card is black” and Event B is “the card is a king,” does the equation hold?

9) Explain or show your reasoning.

10) If Event A is “the card is a face card” and Event B is “the card is a spade,” does the equation hold?

11) Explain or show your reasoning.

##### Coin and Cube

A coin is flipped, then a standard number cube is rolled. Let A represent the event “the coin lands showing heads” and B represent “the standard number cube lands showing 4.”

12) Are events A and B independent or dependent?

dependent

independent

there is not enough information

13) Explain your reasoning.

Find the probabilities. Write answers as fractions in lowest terms.

14) $P\text{(A)}$

15) $P\text{(B)}$

16) $P\text{(A | B)}$

17) $P\text{(B | A)}$

Find the probabilities. Write answers as fractions in lowest terms.

Describe the meaning of the events "not A" and "not B" in each situation, then find the probability.

$P\text{(A | not B)}$

18) Meaning:

19) Probability:

$P\text{(B | not A)}$

20) Meaning:

21) Probability:

22) Select $\textbf{all}$ true statements. Write each corresponding letter in the answer box and separate letters with commas.

a) $P\text{(A)} = P\text{(B)}$ $\quad\quad$ b) $P\text{(A)} = P\text{(A | B)} = P\text{(A | not B)}$ $\quad\quad$ c) $P\text{(not A)} = P\text{(not B)}$ $\quad\quad$ d) $P\text{(B)} = P\text{(B | A)} = P\text{(B | not A)}$

23) Explain your reasoning.