Probability

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Quick recall

• Favorable ÷ total assumes equally likely outcomes.
• Multiply probabilities only when the events are independent.

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Formula Cards

1. Equally likely outcomes

$$P(A)=\frac{\text{Favorable outcomes}}{\text{Total outcomes}}$$

2. Probability range

$$0\le P(A)\le1$$

0 = impossible; 1 = certain.

3. Complement

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

4. Independent events — both

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

Requires independence.

5. Mutually exclusive events — either

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

No overlap.

6. General addition rule

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

Subtract overlap once.

Worked Examples

Example 1

Question: Find the probability of rolling a 4 on a fair six-sided die.

Step 1: Count favorable and total outcomes.

$$\text{Favorable}=1,\qquad\text{Total}=6$$

Step 2: Divide.

$$P(4)=\frac{1}{6}$$

Answer: 1/6.

Example 2

Question: Find the probability of two heads in two independent tosses of a fair coin.

Step 1: Identify each probability.

$$P(H)=\frac{1}{2}$$

Step 2: Multiply for independent events.

$$P(HH)=\frac{1}{2}\times\frac{1}{2}=\frac{1}{4}$$

Answer: 1/4.

Example 3

Question: A bag contains 4 red and 6 blue balls. Find the probability of drawing red at random.

Step 1: Find the total number of balls.

$$4+6=10$$

Step 2: Divide red balls by the total.

$$P(\text{red})=\frac{4}{10}=\frac{2}{5}$$

Answer: 2/5.

Example 4

Question: From a standard 52-card deck, find the probability of drawing a king or a heart.

Step 1: Count each category and the overlap.

$$\text{Kings}=4,\quad\text{Hearts}=13,\quad\text{King of hearts}=1$$

Step 2: Subtract the overlap once.

$$P=\frac{4+13-1}{52}=\frac{16}{52}$$

Step 3: Simplify.

$$P=\frac{4}{13}$$

Answer: 4/13.

Drill

Try each question before opening its answer.

Question 1

Probability of NOT rolling a 6 on a die?

• Show answer

  5/6

Question 2

A bag has 3 red, 5 blue, 2 green balls. Probability of picking green?

• Show answer

  1/5

Question 3

Probability of getting exactly one head in 2 coin tosses.

• Show answer

  1/2

Question 4

Probability of drawing an ace from a deck of 52.

• Show answer

  1/13

Question 5

Two dice are rolled. Probability sum = 7.

• Show answer

  1/6

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