Probability and Conditional Probability

Probability and Conditional Probability

SAT Math — Problem-Solving and Data Analysis

Learning Objectives

By the end of this lesson, you should be able to:

• Calculate probabilities using favorable and total outcomes.

• Express probabilities as fractions, decimals, and percentages.

• Understand complementary events.

• Find probabilities using tables and descriptions of situations.

• Interpret joint probabilities and conditional probabilities.

• Calculate probabilities from two-way frequency tables.

• Identify independent and dependent events.

• Solve probability problems involving multiple events.

• Recognize common SAT probability traps.

1. Understanding Probability

Probability measures how likely an event is to occur.

An event is a particular outcome or collection of outcomes.

For example, rolling an even number on a standard six-sided die is an event.

When all individual outcomes are equally likely, probability is calculated using:

P(Event) = Number of favorable outcomes / Total number of possible outcomes

Suppose a bag contains 5 red balls and 3 blue balls.

Total balls = 5 + 3 = 8

Probability of choosing a red ball:

P(Red) = 5/8

P(Red) = 0.625

P(Red) = 62.5%

All three forms represent the same probability.

Important: The favorable outcomes must be counted from the same sample space as the total outcomes.

2. The Probability Scale

A probability must lie between 0 and 1, inclusive.

Probability can be expressed as:

• A fraction, such as 3/5

• A decimal, such as 0.6

• A percentage, such as 60%

To convert a fraction to a decimal, divide the numerator by the denominator.

To convert a decimal to a percentage, multiply by 100.

SAT Tip: Always check whether the question asks for a fraction, decimal, percentage, or numerical probability.

3. Sample Spaces and Equally Likely Outcomes

A sample space is the collection of all possible outcomes.

For a fair six-sided die:

Sample space = {1, 2, 3, 4, 5, 6}

The probability of rolling an even number is:

Favorable outcomes = {2, 4, 6}

Number of favorable outcomes = 3

Total outcomes = 6

P(Even) = 3/6 = 1/2

Answer: 1/2

The number of outcomes is not always the same as their probabilities. The favorable-over-total formula applies directly when all the individual outcomes being counted are equally likely.

4. Complementary Events

The complement of an event means that the event does not occur.

The probability rule is:

P(Not A) = 1 − P(A)

Example:

A machine has a 0.08 probability of producing a defective product.

What is the probability that a randomly selected product is not defective?

P(Not defective) = 1 − 0.08

P(Not defective) = 0.92

Answer: 0.92 or 92%

Complementary events are especially useful when a problem asks for the probability that something does not happen.

5. Probability from Frequency Tables

SAT questions frequently provide data in tables.

Consider the following school club survey:

Suppose a student is selected randomly from all 100 students.

Example A — Probability of Grade 11

Number of Grade 11 students = 50

Total students = 100

P(Grade 11) = 50/100 = 1/2

Example B — Probability of Playing Sports

Students who play sports = 54

Total students = 100

P(Plays Sports) = 54/100 = 0.54

Example C — Probability of Grade 11 and Playing Sports

Students who are both in Grade 11 and play sports = 30

Total students = 100

P(Grade 11 and Plays Sports) = 30/100 = 0.30

Notice that the denominator depends on the group from which the student is selected.

6. Joint Probability

Joint probability is the probability that two events both occur.

The word AND is the key signal.

Example:

Using the school survey, what is the probability that a randomly selected student is both in Grade 12 AND plays sports?

Grade 12 students who play sports = 24

Total students = 100

P(Grade 12 and Sports) = 24/100

P(Grade 12 and Sports) = 0.24

Answer: 24%

For a joint probability from a two-way table, use the cell representing both conditions and divide by the total for the full sample space.

7. Conditional Probability

Conditional probability means finding the probability of an event given that another condition is already known to be true.

Words such as these often indicate conditional probability:

• Given that

• Among

• Of those who

• If a student is selected from

• Knowing that

The conditional probability formula is:

P(A | B) = P(A and B) / P(B)

Read P(A | B) as:

"The probability of A given B."

For frequency tables:

Conditional probability = Number satisfying both conditions / Number satisfying the given condition

Example

Using the school survey:

A student is selected randomly from Grade 11.

What is the probability that this student plays sports?

Step 1: Identify the condition.

The student is already known to be in Grade 11.

Step 2: Identify the new total.

There are 50 Grade 11 students.

Step 3: Identify favorable outcomes within Grade 11.

30 Grade 11 students play sports.

Step 4: Calculate.

P(Sports | Grade 11) = 30/50

P(Sports | Grade 11) = 3/5

Answer: 0.6 or 60%

The denominator is 50, NOT 100, because only Grade 11 students are being considered.

8. The Most Important SAT Difference: AND vs. GIVEN

Consider:

100 students are surveyed.

40 are members of the science club.

Of those 40, 25 also play basketball.

Question A

What is the probability that a randomly selected student is in the science club AND plays basketball?

Both conditions = 25 students

Total surveyed = 100

P(Science and Basketball) = 25/100 = 0.25

Question B

Given that a student is in the science club, what is the probability that the student plays basketball?

Students in the science club = 40

Science club students who play basketball = 25

P(Basketball | Science) = 25/40 = 0.625

Important SAT distinction:

AND usually refers to the intersection of events.

GIVEN THAT restricts the sample space to a particular group.

This difference determines the denominator.

9. Two-Way Tables and Conditional Probability

Suppose a company surveys 200 customers.

Question 1

What is the probability that a randomly selected customer shops online?

P(Online) = 130/200 = 0.65

Question 2

What is the probability that a randomly selected customer is a teenager AND shops online?

P(Teenager and Online) = 60/200 = 0.30

Question 3

Given that a customer is a teenager, what is the probability that the customer shops online?

P(Online | Teenager) = 60/80 = 0.75

Question 4

Given that a customer shops online, what is the probability that the customer is a teenager?

P(Teenager | Online) = 60/130 = 6/13

SAT Tip: P(A | B) and P(B | A) are generally not equal.

Always identify the condition before selecting the denominator.

10. OR Probability

The word OR means an event belongs to at least one of two categories.

The general formula is:

P(A or B) = P(A) + P(B) − P(A and B)

The intersection is subtracted because it was counted twice.

Example:

In a group of 100 students:

45 study French.

35 study Spanish.

15 study both.

Find the probability that a randomly selected student studies French or Spanish.

P(French or Spanish) = (45 + 35 − 15)/100

P(French or Spanish) = 65/100

Answer: 0.65

Mutually Exclusive Events

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

For mutually exclusive events:

P(A and B) = 0

Therefore:

P(A or B) = P(A) + P(B)

Example:

On one roll of a die, rolling a 2 and rolling a 5 are mutually exclusive events.

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

11. Independent and Dependent Events

Two events are independent when knowing that one occurred does not change the probability of the other.

For independent events:

P(A and B) = P(A) × P(B)

Example:

A fair coin is tossed twice.

Probability of heads on the first toss = 1/2

Probability of heads on the second toss = 1/2

P(Two Heads) = (1/2)(1/2)

Answer: 1/4

Dependent Events

Events are dependent when the outcome of one event changes the probability of another.

Example:

A bag contains 4 red balls and 3 blue balls.

Two balls are drawn without replacement.

Find the probability that both are red.

Step 1: Probability the first ball is red.

P(First red) = 4/7

Step 2: After removing one red ball, 3 red balls remain among 6 balls.

P(Second red | First red) = 3/6

Step 3: Multiply.

P(Both red) = (4/7)(3/6)

P(Both red) = 12/42 = 2/7

Answer: 2/7

For events that are not necessarily independent, the multiplication formula is:

P(A and B) = P(A) × P(B | A)

12. Probability With and Without Replacement

With Replacement

An item is selected and then returned before the next selection.

The total number of items remains unchanged.

For repeated independent selections, multiply the same probabilities when appropriate.

Without Replacement

The selected item is not returned.

The total number of items decreases after selection.

Example:

A box contains 5 blue balls and 3 red balls.

Two balls are drawn without replacement.

Find the probability that both are blue.

First draw = 5/8

Second draw after a blue ball = 4/7

P(Both blue) = (5/8)(4/7)

P(Both blue) = 20/56 = 5/14

Answer: 5/14

13. Relative Frequency and Experimental Probability

Experimental probability is estimated from actual observations.

Experimental probability = Number of times event occurred / Total number of trials

Example:

A basketball player makes 42 successful shots in 60 attempts.

Estimated probability of success:

42/60 = 0.7

Answer: 70%

This is an estimate based on recorded performance.

It does not guarantee that exactly 70% of future shots will be successful.

14. Worked SAT-Style Examples

Example 1 — Basic Probability

Problem: A bag contains 6 green, 5 blue, and 4 yellow marbles. What is the probability of selecting a blue marble?

Solution:

Total marbles = 6 + 5 + 4 = 15

Blue marbles = 5

P(Blue) = 5/15

P(Blue) = 1/3

Answer: 1/3

Example 2 — Complement

Problem: The probability that a machine produces a defective item is 0.035. What is the probability that an item is not defective?

Solution:

P(Not defective) = 1 − 0.035

P(Not defective) = 0.965

Answer: 0.965

Example 3 — Conditional Probability

Problem: Among 80 students, 50 study mathematics, and 30 of those mathematics students also study physics. Given that a student studies mathematics, what is the probability that the student also studies physics?

Solution:

Given group = 50 mathematics students

Favorable outcomes = 30 students studying both subjects

P(Physics | Mathematics) = 30/50

P(Physics | Mathematics) = 3/5

Answer: 0.6

Example 4 — Joint Probability

Problem: In a survey of 120 students, 35 are in Grade 11 and participate in the debate club. What is the probability that a randomly selected student satisfies both conditions?

Solution:

Students satisfying both conditions = 35

Total students = 120

P(Both) = 35/120

P(Both) = 7/24

Answer: 7/24

Example 5 — OR Probability

Problem: Among 200 students, 80 study French, 70 study Spanish, and 25 study both. What is the probability that a student studies at least one of these languages?

Solution:

Students studying at least one = 80 + 70 − 25

= 125

Probability = 125/200

Answer: 0.625

Example 6 — Without Replacement

Problem: A bag contains 5 red and 4 blue balls. Two balls are drawn without replacement. What is the probability that both are red?

Solution:

First red = 5/9

Second red given first red = 4/8

P(Both red) = (5/9)(4/8)

P(Both red) = 20/72

Answer: 5/18

Example 7 — Reverse Conditional Probability

Problem: Of 150 students, 60 participate in a science club. Among those 60, 36 are Grade 12 students. What is the probability that a randomly selected science club member is in Grade 12?

Solution:

Given condition = science club membership

Total science club members = 60

Grade 12 science club members = 36

P(Grade 12 | Science Club) = 36/60

Answer: 0.6

15. Common SAT Mistakes

• Using the total population as the denominator when the question specifies a subgroup.

• Confusing "A and B" with "A given B."

• Assuming P(A | B) equals P(B | A).

• Forgetting to subtract the overlap when calculating OR probability.

• Adding probabilities when the problem requires multiplication.

• Assuming independent events when objects are selected without replacement.

• Forgetting that the total number of objects changes after a draw without replacement.

• Confusing a probability of 0.25 with 25 instead of 25%.

• Treating experimental probability as a guaranteed future result.

• Misreading row and column totals in two-way tables.

• Using favorable outcomes from the wrong group.

• Counting overlapping categories twice.

• Forgetting to simplify fractions where helpful.

• Assuming mutually exclusive events are automatically independent.

• Failing to identify whether a question asks for a decimal, fraction, or percentage.

16. Final SAT Revision Checklist

Before attempting the practice questions, make sure you can:

• Calculate simple probabilities.

• Express probabilities as fractions, decimals, and percentages.

• Use complementary probability.

• Read frequency tables correctly.

• Calculate joint probabilities using AND.

• Calculate conditional probabilities using GIVEN THAT.

• Solve OR probability questions with overlapping groups.

• Distinguish dependent and independent events.

• Calculate probabilities with and without replacement.

• Interpret relative frequencies and experimental probability.

Final SAT Strategy: First identify the group from which the selection is made. Then count the favorable outcomes within that group. The most common error in conditional probability is choosing the wrong denominator.