SAT — mathFoundational3 min read

Basic Probability

Master probability fundamentals for the Digital SAT. Calculate P(event), complementary probability, and independent events.

Tutor AI Team
Tutor AI Team

Probability measures how likely an event is to occur, on a scale from 0 (impossible) to 1 (certain). The Digital SAT tests basic probability through straightforward calculations, complementary events, and independent events.

Core Concepts

Probability of an Event

P(event)=number of favourable outcomestotal number of outcomesP(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}

Example: A bag has 3 red, 4 blue, and 5 green marbles. P(red)=312=14P(\text{red}) = \frac{3}{12} = \frac{1}{4}.

Complementary Events

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

If P(rain)=0.3P(\text{rain}) = 0.3, then P(no rain)=0.7P(\text{no rain}) = 0.7.

"Or" — Addition Rule

For mutually exclusive events (can't happen at the same time): P(A or B)=P(A)+P(B)P(A \text{ or } B) = P(A) + P(B)

For events that CAN overlap: P(A or B)=P(A)+P(B)−P(A and B)P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)

"And" — Multiplication Rule

For independent events (one doesn't affect the other): P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)

Example: Flip a coin and roll a die. P(heads and 6)=12×16=112P(\text{heads and 6}) = \frac{1}{2} \times \frac{1}{6} = \frac{1}{12}.

Probability from Tables

The SAT often presents data in tables and asks for the probability of selecting a person/item with a given characteristic.

Strategy Tips

Tip 1: Favourable Over Total

Always set up the fraction: favourable outcomes / total outcomes.

Tip 2: Use Complementary Probability

If "at least one" is hard to calculate directly, use P(at least one)=1−P(none)P(\text{at least one}) = 1 - P(\text{none}).

Tip 3: Read the Denominator Carefully

Is the question about the entire population or a subgroup? This changes the denominator.

Worked Example: Example 1

Problem

A jar has 5 red and 7 blue marbles. What is the probability of picking a blue marble?

P=712P = \frac{7}{12}

Solution

Worked Example: Example 2

Problem

Roll a standard die. What is P(not 3)P(\text{not 3})?

P=1−16=56P = 1 - \frac{1}{6} = \frac{5}{6}

Solution

Worked Example: SAT-Style

Problem

A survey of 200 students found 80 play sports and 60 are in band. 20 do both. What is P(sports or band)P(\text{sports or band})?

P=80+60−20200=120200=0.6P = \frac{80 + 60 - 20}{200} = \frac{120}{200} = 0.6

Solution

Worked Example: Example 4

Problem

Two independent events: P(A)=0.4P(A) = 0.4 and P(B)=0.3P(B) = 0.3. Find P(A and B)P(A \text{ and } B).

P=0.4×0.3=0.12P = 0.4 \times 0.3 = 0.12

Solution

Worked Example: Example 5

Problem

A coin is flipped 3 times. What is the probability of getting at least one head?

P(no heads)=(0.5)3=0.125P(\text{no heads}) = (0.5)^3 = 0.125

P(at least one head)=1−0.125=0.875P(\text{at least one head}) = 1 - 0.125 = 0.875

Solution

Practice Problems

  1. Problem 1

    A deck of 52 cards. PP(drawing a heart)?

    Problem 2

    PP(rolling 2 or 5 on a die)?

    Problem 3

    Two coins flipped. PP(both heads)?

    Problem 4

    Of 150 people, 90 like coffee, 60 like tea, 30 like both. PP(likes coffee or tea)?

    Problem 5

    PP(not rain) = 0.65. PP(rain)?

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Common Mistakes

  • Using the wrong denominator. "Of the students who play sports..." means the denominator is sports players, not all students.
  • Adding probabilities of non-mutually-exclusive events without subtracting overlap. Always subtract P(A and B)P(A \text{ and } B).
  • Confusing independent with mutually exclusive. Independent: outcomes don't affect each other. Mutually exclusive: can't happen simultaneously.

Key Takeaways

  • ✓

    P(event)=favourabletotalP(\text{event}) = \frac{\text{favourable}}{\text{total}}, always between 0 and 1.

  • ✓

    Complementary: P(not A)=1−P(A)P(\text{not } A) = 1 - P(A).

  • ✓

    Or (mutually exclusive): add. Or (overlap): add and subtract overlap.

  • ✓

    And (independent): multiply.

  • ✓

    "At least one": use 1−P(none)1 - P(\text{none}).

  • ✓

    Read the denominator carefully — is it the total or a subgroup?

Tags:
#sat#math#problem-solving#probability

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