Complete lesson: probability rules, two-way tables, conditional probability, and 10 practice questions.
📖 What Is Probability?
Probability measures the likelihood of an event occurring. It is always a number between 0 and 1.
\( P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \)
Probability
\(P(E) = \frac{f}{n}\)
0 = impossible, 1 = certain
Complement
\(P(\text{not E}) = 1 - P(E)\)
Probability of event NOT happening
Conditional Probability
\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)
Probability of A given B
Two-Way Table
Frequency table
Shows data classified by two variables
1. Basic Probability
📌 RULE: Probability
\[
P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}
\]
Example: What is the probability of rolling a 4 on a standard die?
\[
P(4) = \frac{1}{6}
\]
💡 Strategy — Probability
1. Count the total number of possible outcomes.
2. Count the number of favorable outcomes.
3. Divide: favorable ÷ total.
4. Express as a fraction, decimal, or percentage.
📝 SOLVED EXAMPLE 1 — Basic Probability
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing a blue marble?
💡 Tip: Use the column/row totals as the "total" in the denominator.
3. Conditional Probability
📌 RULE: Conditional Probability
\[
P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\text{Number in both A and B}}{\text{Total in B}}
\]
This is the probability of event A occurring, given that event B has already occurred.
Read as: "Probability of A given B"
💡 Strategy — Conditional Probability
1. Identify what the condition is (the "given" part).
2. Find the total for that condition (this is your denominator).
3. Find the number that satisfies both conditions (your numerator).
4. Divide.
📝 SOLVED EXAMPLE 3 — Conditional Probability
Using the table above, what is the probability that a student is a freshman, given that the student is female?
Step 1: Condition: "given female" → total female = 150
💡 Tip: The probability of "not" is always 1 minus the probability of "yes."
🧪 Practice Questions
Solve each problem using the rules above. Click "Show Answer" to see the full solution.
Question 1
A bag contains 4 red, 6 blue, and 10 green marbles. What is the probability of drawing a green marble?
A) \(\frac{1}{4}\)
B) \(\frac{1}{2}\)
C) \(\frac{3}{4}\)
D) \(\frac{2}{5}\)
✓ Answer: B
Total = 4+6+10 = 20. Green = 10. \(P = 10/20 = 1/2\)
📝 Solution: \(10/20 = 1/2\)
Question 2
A standard deck has 52 cards. What is the probability of drawing an Ace (4 aces in a deck)?
A) \(\frac{1}{52}\)
B) \(\frac{1}{26}\)
C) \(\frac{1}{13}\)
D) \(\frac{4}{13}\)
✓ Answer: C
\(P(\text{Ace}) = 4/52 = 1/13\)
📝 Solution: \(4/52 = 1/13\)
Question 3
Using the table below, what is the probability that a randomly chosen student is male?
Gender
Freshman
Sophomore
Total
Male
30
20
50
Female
40
30
70
Total
70
50
120
A) \(\frac{5}{12}\)
B) \(\frac{5}{12}\)
C) \(\frac{7}{12}\)
D) \(\frac{1}{2}\)
✓ Answer: B
Male total = 50. Grand total = 120. \(P = 50/120 = 5/12\)
📝 Solution: \(50/120 = 5/12\)
Question 4
Using the table above, what is the probability that a student is female, given that they are a freshman?
A) \(\frac{1}{3}\)
B) \(\frac{2}{5}\)
C) \(\frac{4}{7}\)
D) \(\frac{3}{5}\)
✓ Answer: C
Condition: Freshman. Total Freshman = 70. Female AND Freshman = 40. \(P = 40/70 = 4/7\)
📝 Solution: \(40/70 = 4/7\)
Question 5
If \(P(\text{Event}) = 0.35\), what is \(P(\text{not Event})\)?
A) 0.35
B) 0.45
C) 0.55
D) 0.65
✓ Answer: D
\(P(\text{not}) = 1 - 0.35 = 0.65\)
📝 Solution: \(1 - 0.35 = 0.65\)
Question 6
A jar contains 5 red, 3 yellow, and 2 blue marbles. What is the probability of drawing a yellow marble?
A) \(\frac{1}{10}\)
B) \(\frac{1}{5}\)
C) \(\frac{1}{4}\)
D) \(\frac{3}{10}\)
✓ Answer: D
Total = 5+3+2 = 10. Yellow = 3. \(P = 3/10\)
📝 Solution: \(3/10\)
Question 7
Using the table above, what is the probability that a randomly chosen student is a sophomore?
A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{5}{12}\)
D) \(\frac{7}{12}\)
✓ Answer: C
Sophomore total = 50. Grand total = 120. \(P = 50/120 = 5/12\)
📝 Solution: \(50/120 = 5/12\)
Question 8
If \(P(\text{rain}) = 0.2\), what is the probability that it does NOT rain?
A) 0.2
B) 0.8
C) 0.5
D) 1.0
✓ Answer: B
\(P(\text{not rain}) = 1 - 0.2 = 0.8\)
📝 Solution: \(1 - 0.2 = 0.8\)
Question 9
Using the table above, what is the probability that a student is male, given that they are a sophomore?
A) \(\frac{2}{5}\)
B) \(\frac{1}{2}\)
C) \(\frac{3}{5}\)
D) \(\frac{2}{3}\)
✓ Answer: A
Condition: Sophomore. Total Sophomore = 50. Male AND Sophomore = 20. \(P = 20/50 = 2/5\)
📝 Solution: \(20/50 = 2/5\)
Question 10
A spinner has 8 equal sections: 3 red, 2 blue, and 3 green. What is the probability of spinning green?
A) \(\frac{1}{4}\)
B) \(\frac{1}{3}\)
C) \(\frac{3}{8}\)
D) \(\frac{3}{5}\)
✓ Answer: C
Total sections = 8. Green = 3. \(P = 3/8\)
📝 Solution: \(3/8\)
🎉 WELL DONE!
You've completed the Probability & Conditional Probability lesson. You now know how to calculate basic probability, use two-way tables, and find conditional probabilities.