Training Probability & Combinatorics Probability Fundamentals
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Probability Fundamentals

24 min Probability & Combinatorics

Probability Fundamentals

Probability of an Event

If all outcomes in a sample space $S$ are equally likely:

$$P(A) = \frac{|A|}{|S|} = \frac{\text{favorable outcomes}}{\text{total outcomes}}$$

$0 \leq P(A) \leq 1$ for any event $A$.

Complement

$$P(A^c) = 1 - P(A)$$

Addition Rule

$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$

If $A$ and $B$ are mutually exclusive: $P(A \cup B) = P(A) + P(B)$.

Multiplication Rule

$$P(A \cap B) = P(A) \cdot P(B|A)$$

If $A$ and $B$ are independent: $P(A \cap B) = P(A) \cdot P(B)$.

Example 1

A fair die is rolled. What is $P(\text{even})$?

Even outcomes: $\{2, 4, 6\}$, total: 6.

$$P(\text{even}) = \frac{3}{6} = \frac{1}{2}$$

Example 2

A card is drawn from a standard 52-card deck. What is $P(\text{heart or face card})$?

Hearts = 13, face cards = 12, heart face cards = 3.

$$P = \frac{13 + 12 - 3}{52} = \frac{22}{52} = \frac{11}{26}$$

Example 3

Two dice are rolled. What is $P(\text{sum} = 7)$?

Favorable: $(1,6),(2,5),(3,4),(4,3),(5,2),(6,1) = 6$ outcomes. Total: $36$.

$$P = \frac{6}{36} = \frac{1}{6}$$

Practice Problems

1. A fair coin is flipped 3 times. $P(\text{exactly 2 heads})$?
2. A bag has 5 red, 3 blue, 2 green marbles. $P(\text{red})$?
3. $P(\text{not red})$ from the same bag?
4. Two cards drawn without replacement. $P(\text{both aces})$?
5. $P(\text{rolling a 5 or 6 on a die})$?
6. $P(\text{sum of 2 dice} \leq 4)$?
7. A box has 8 good and 2 defective items. Pick 1 at random. $P(\text{good})$?
8. Are flipping a coin and rolling a die independent?
9. $P(\text{at least one head in 3 flips})$?
10. A jar has 4 red, 6 blue. Two drawn without replacement. $P(\text{both blue})$?
11. What does mutually exclusive mean?
12. If $P(A) = 0.3$ and $P(B) = 0.5$ and $A,B$ independent, find $P(A \cap B)$.
Show Answer Key

1. $\binom{3}{2}/2^3 = 3/8$

2. $5/10 = 1/2$

3. $1 - 1/2 = 1/2$

4. $\frac{4}{52} \cdot \frac{3}{51} = \frac{12}{2652} = \frac{1}{221}$

5. $2/6 = 1/3$

6. Outcomes $\leq 4$: $(1,1),(1,2),(1,3),(2,1),(2,2),(3,1) = 6$; $P = 6/36 = 1/6$

7. $8/10 = 4/5$

8. Yes, the outcomes do not affect each other

9. $1 - P(\text{no heads}) = 1 - 1/8 = 7/8$

10. $\frac{6}{10} \cdot \frac{5}{9} = \frac{30}{90} = 1/3$

11. They cannot both occur at the same time

12. $0.3 \times 0.5 = 0.15$

Combinations & the Binomial Theorem Conditional Probability & Expected Value