Probability

Quick Answer: Probability questions test your ability to calculate the likelihood of events. Know the basic formula (favorable ÷ total), rules for compound events (multiply for "and," add for "or" with mutually exclusive events), and expected value calculations. These concepts appear within the Data Interpretation section.

Basic Probability

The Probability Formula

$P(\text{event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$

Probability Scale

Worked Example: Basic Probability

Problem: A bag contains 4 red marbles, 6 blue marbles, and 2 green marbles. If you draw one marble at random, what is the probability of drawing a blue marble?

Solution:

• Favorable outcomes: 6 (blue marbles)
• Total outcomes: 4 + 6 + 2 = 12 (all marbles)
• P(blue) = 6/12 = 1/2 or 0.5 or 50%

Complement Rule

The complement of an event is everything that is NOT that event.

$P(\text{not A}) = 1 - P(\text{A})$

Example: If P(rain) = 0.3, then P(no rain) = 1 - 0.3 = 0.7

Test Tip: When a problem asks for the probability of something NOT happening, use the complement rule. It's often easier than counting all non-favorable outcomes.

Compound Events

Independent Events

Events are independent if one event doesn't affect the probability of the other.

For independent events (both happening): $P(\text{A and B}) = P(\text{A}) \times P(\text{B})$

Worked Example: Independent Events

Problem: A fair coin is flipped twice. What is the probability of getting heads both times?

Solution:

• P(heads on first flip) = 1/2
• P(heads on second flip) = 1/2
• P(heads and heads) = 1/2 × 1/2 = 1/4 or 0.25

Worked Example: Multiple Independent Events

Problem: A spinner has 4 equal sections: red, blue, green, yellow. If you spin three times, what is the probability of landing on blue all three times?

Solution:

• P(blue on each spin) = 1/4
• P(blue, blue, blue) = 1/4 × 1/4 × 1/4 = 1/64

Mutually Exclusive Events

Events are mutually exclusive if they cannot happen at the same time.

For mutually exclusive events (either happening): $P(\text{A or B}) = P(\text{A}) + P(\text{B})$

Worked Example: Mutually Exclusive Events

Problem: A standard die is rolled. What is the probability of rolling a 2 or a 5?

Solution:

• P(2) = 1/6
• P(5) = 1/6
• P(2 or 5) = 1/6 + 1/6 = 2/6 = 1/3

Non-Mutually Exclusive Events

When events CAN happen together, subtract the overlap:

$P(\text{A or B}) = P(\text{A}) + P(\text{B}) - P(\text{A and B})$

Worked Example: Non-Mutually Exclusive

Problem: In a class of 30 students, 18 play sports, 12 are in band, and 5 do both. What is the probability that a randomly selected student plays sports OR is in band?

Solution:

• P(sports) = 18/30
• P(band) = 12/30
• P(both) = 5/30
• P(sports or band) = 18/30 + 12/30 - 5/30 = 25/30 = 5/6

Dependent Events

Events are dependent when the outcome of one affects the probability of the other.

$P(\text{A and B}) = P(\text{A}) \times P(\text{B given A})$

Worked Example: Dependent Events (Without Replacement)

Problem: A bag contains 5 red and 3 blue marbles. If two marbles are drawn WITHOUT replacement, what is the probability both are red?

Solution:

• P(1st red) = 5/8
• After drawing one red, there are 4 red and 3 blue (7 total)
• P(2nd red | 1st was red) = 4/7
• P(both red) = 5/8 × 4/7 = 20/56 = 5/14

Expected Value

Expected value is the average outcome you'd expect over many trials.

$E = (\text{value}_1 \times P_1) + (\text{value}_2 \times P_2) + \ldots$

Worked Example: Expected Value

Problem: A game costs $2 to play. You roll a die: roll 6 and win $10, roll 1-5 and win nothing. What is the expected value?

Solution:

1. Calculate winnings (net of cost):

   • Roll 6: Win $10 - $2 = $8 net gain
   • Roll 1-5: Win $0 - $2 = -$2 net loss

2. Calculate probabilities:

   • P(6) = 1/6
   • P(1-5) = 5/6

3. Expected value:

   • E = ($8 × 1/6) + (-$2 × 5/6)
   • E = $8/6 - $10/6
   • E = -$2/6 ≈ -$0.33

Interpretation: On average, you lose about 33 cents per game.

Common Probability Scenarios

Coin Flips

• P(heads) = P(tails) = 1/2
• Two flips: HH, HT, TH, TT (4 outcomes)

Dice Rolls

• P(any specific number) = 1/6
• P(even) = P(odd) = 3/6 = 1/2
• P(sum of 7 on two dice) = 6/36 = 1/6

Card Draws (Standard 52-Card Deck)

• P(specific card) = 1/52
• P(any ace) = 4/52 = 1/13
• P(any heart) = 13/52 = 1/4
• P(face card) = 12/52 = 3/13

Calculator Tip: For probability problems, the calculator is useful for converting fractions to decimals (divide numerator by denominator) and for multiplying fractions. But be sure to express your final answer in the format the question requests.