Compound Probability

For Elementary Students

What is Compound Probability?

Compound probability is when you find the chance of two or more things happening together!

Think about it like this: Instead of flipping ONE coin, what if you flip TWO coins? What's the chance you get heads both times? That's compound probability!

Examples of Compound Events

Flipping two coins — What's the chance of heads, then heads?

Rolling two dice — What's the chance of a 6, then another 6?

Picking two cards — What's the chance of getting two aces?

Spinning a spinner twice — What's the chance of landing on red both times?

The Big Rule: Multiply!

When you want BOTH things to happen, you MULTIPLY the probabilities!

Formula: P(A and B) = P(A) × P(B)

Example: Two Coin Flips

Question: What's the chance of flipping heads TWICE?

Step 1: First flip

• Chance of heads = 1/2

Step 2: Second flip

• Chance of heads = 1/2

Step 3: Multiply for "both"

1/2 × 1/2 = 1/4

Answer: 1/4 or 25%

There's only 1 way to get HH out of 4 possible outcomes (HH, HT, TH, TT)!

Example: Rolling a Die and Flipping a Coin

Question: What's the chance of rolling a 5 AND flipping heads?

Step 1: Roll a 5

• Chance = 1/6

Step 2: Flip heads

• Chance = 1/2

Step 3: Multiply

1/6 × 1/2 = 1/12

Answer: 1/12

Independent Events

Independent means one event doesn't change the other!

Examples:

• Flipping two different coins (first flip doesn't affect second)
• Rolling two dice (first roll doesn't affect second)
• Spinning a spinner twice (first spin doesn't affect second)

For independent events: Just multiply the probabilities!

Dependent Events

Dependent means the first event CHANGES the second!

Examples:

• Pick a marble, DON'T put it back, pick another (fewer marbles left!)
• Draw a card, DON'T replace it, draw another (fewer cards left!)

Example: Picking Marbles (Dependent)

Bag has: 3 red marbles, 2 blue marbles (5 total)

Question: Pick red, then pick blue (WITHOUT putting red back)

Step 1: First pick (red)

3 red out of 5 marbles = 3/5

Step 2: Second pick (blue)

Now only 4 marbles left!
Still 2 blue
Chance = 2/4 = 1/2

Step 3: Multiply

3/5 × 1/2 = 3/10

Answer: 3/10 or 30%

With Replacement vs. Without

With replacement = Put it back → Independent (same chances every time)

Without replacement = Don't put it back → Dependent (chances change!)

Tree Diagrams

Tree diagrams help you see ALL the possible outcomes!

Example: Flip a coin twice

First Flip    Second Flip    Outcome
    H --------- H --------- HH (1/2 × 1/2 = 1/4)
   /
Start
   \
    T --------- H --------- TH (1/2 × 1/2 = 1/4)
        \
         T ----- HT -------- (1/2 × 1/2 = 1/4)
          \
           T ---- TT -------- (1/2 × 1/2 = 1/4)

Four possible outcomes, each with probability 1/4!

"And" vs. "Or"

"AND" (both happen): MULTIPLY

• What's the chance of heads AND heads? → 1/2 × 1/2 = 1/4

"OR" (either happens): ADD

• What's the chance of rolling a 3 OR a 4? → 1/6 + 1/6 = 2/6 = 1/3

Real-Life Examples

Weather: "What's the chance it rains Saturday AND Sunday?"

• If each day has 50% rain: 0.5 × 0.5 = 0.25 = 25%

Games: "What's the chance of drawing two hearts from a deck?"

• Depends if you put the first card back!

Quick Tips

Tip 1: "And" means multiply

Tip 2: "Or" means add (if they can't both happen)

Tip 3: No replacement? Second probability changes!

Tip 4: Draw a tree diagram if you're stuck!

For Junior High Students

Formal Definition

Compound event: An event consisting of two or more simple events

Compound probability: The probability that multiple events occur together or in sequence

Notation: P(A and B) or P(A ∩ B)

Independent Events

Definition: Events where the occurrence of one does NOT affect the probability of the other

Mathematical test: Events A and B are independent if:

P(A and B) = P(A) × P(B)

Examples:

• Flipping multiple coins
• Rolling multiple dice
• Sampling with replacement

Multiplication Rule for Independent Events

Formula: P(A and B) = P(A) × P(B)

Extends to multiple events: P(A and B and C) = P(A) × P(B) × P(C)

Example 1: Two Coin Flips

Event: Get heads twice in a row

Calculate:

P(H on first flip) = 1/2
P(H on second flip) = 1/2

P(HH) = 1/2 × 1/2 = 1/4

Answer: 1/4 or 25%

Example 2: Rolling Two Dice

Event: Roll 6 on first die AND 6 on second die

Calculate:

P(6 on die 1) = 1/6
P(6 on die 2) = 1/6

P(both 6s) = 1/6 × 1/6 = 1/36

Answer: 1/36 ≈ 2.78%

Example 3: Three Independent Events

Event: Flip heads three times in a row

Calculate:

P(HHH) = 1/2 × 1/2 × 1/2 = 1/8

Answer: 1/8 = 12.5%

Dependent Events

Definition: Events where the occurrence of one AFFECTS the probability of the other

Formula: P(A and B) = P(A) × P(B|A)

Where P(B|A) = "probability of B given that A has occurred"

Examples:

• Sampling without replacement
• Drawing cards without replacing
• Sequential selections from finite populations

Example 1: Drawing Cards (No Replacement)

Standard deck: 52 cards, 4 aces

Event: Draw ace, then draw another ace (no replacement)

Calculate:

P(first ace) = 4/52 = 1/13

After removing one ace:
P(second ace | first was ace) = 3/51

P(two aces) = 4/52 × 3/51 = 12/2652 = 1/221

Answer: 1/221 ≈ 0.45%

Example 2: Marbles from Bag

Bag: 5 red, 3 blue marbles (8 total)

Event: Pick red, then pick blue (no replacement)

Calculate:

P(red first) = 5/8

After removing red marble:
P(blue second | red first) = 3/7

P(red then blue) = 5/8 × 3/7 = 15/56

Answer: 15/56 ≈ 26.8%

Example 3: Same Color (Dependent)

Same bag: 5 red, 3 blue

Event: Pick red, then pick red again (no replacement)

Calculate:

P(red first) = 5/8
P(red second | red first) = 4/7

P(two red) = 5/8 × 4/7 = 20/56 = 5/14

Answer: 5/14 ≈ 35.7%

With Replacement vs. Without Replacement

With replacement: Item returned before next selection

• Events become independent
• Probabilities remain constant

Without replacement: Item not returned

• Events become dependent
• Probabilities change based on previous outcomes

Example: Compare Methods

Bag: 4 green, 6 yellow balls (10 total)

Event: Pick green twice

WITH replacement:

P(G) = 4/10 each time
P(GG) = 4/10 × 4/10 = 16/100 = 4/25

WITHOUT replacement:

P(first G) = 4/10
P(second G | first G) = 3/9 = 1/3
P(GG) = 4/10 × 1/3 = 4/30 = 2/15

Results differ: 4/25 ≈ 16% vs 2/15 ≈ 13.3%

Tree Diagrams

Purpose: Visual representation of all possible outcomes and their probabilities

Method:

1. Draw branches for each outcome of first event
2. From each branch, draw branches for second event
3. Multiply probabilities along each path
4. Sum of all path probabilities = 1

Example: Dependent Events Tree

Bag: 2 red (R), 3 blue (B), draw 2 without replacement

First       Second      Outcome    Probability
  R (2/5) --- R (1/4) --- RR ------- 2/5 × 1/4 = 2/20
        \
         B (3/4) --- RB ------- 2/5 × 3/4 = 6/20

  B (3/5) --- R (2/4) --- BR ------- 3/5 × 2/4 = 6/20
        \
         B (2/4) --- BB ------- 3/5 × 2/4 = 6/20

Check: 2/20 + 6/20 + 6/20 + 6/20 = 20/20 = 1 ✓

"And" vs. "Or" Probabilities

P(A and B): Both events occur → Multiply (if independent)

P(A ∩ B) = P(A) × P(B)

P(A or B): At least one occurs → Add (if mutually exclusive)

P(A ∪ B) = P(A) + P(B)  [if mutually exclusive]
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)  [general formula]

Example: "Or" Probability

Roll a die. Find P(3 or 4)

Events are mutually exclusive (can't roll both simultaneously)

P(3) = 1/6
P(4) = 1/6
P(3 or 4) = 1/6 + 1/6 = 2/6 = 1/3

"At Least One" Problems

Strategy: Use complement

P(at least one) = 1 − P(none)

Easier than listing all cases!

Example: At Least One Success

Flip coin three times. P(at least one heads)?

Method 1: Complement

P(no heads) = P(TTT) = 1/2 × 1/2 × 1/2 = 1/8
P(at least one H) = 1 - 1/8 = 7/8

Method 2: List all (longer)

HHH, HHT, HTH, HTT, THH, THT, TTH (7 out of 8)
P = 7/8

Both give 7/8 = 87.5%

Conditional Probability

P(B|A): Probability of B given that A has already occurred

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

Rearranging: P(A and B) = P(A) × P(B|A)

This is the general multiplication rule!

Example: Conditional Probability

Deck: 52 cards

Given: First card was ace

Find: P(second card is ace | first was ace)

Calculate:

After removing one ace:
3 aces left, 51 cards left
P(second ace | first ace) = 3/51 = 1/17

Applications

Quality Control: P(two defective items in sample)

Medical Testing: P(positive test and disease)

Weather Forecasting: P(rain multiple consecutive days)

Genetics: P(offspring inheriting specific traits)

Card Games: P(specific poker hands)

Sports: P(team winning multiple games)

Common Errors

Error 1: Adding instead of multiplying for "and"

❌ P(H and H) = 1/2 + 1/2 = 1
✓ P(H and H) = 1/2 × 1/2 = 1/4

Error 2: Not adjusting for dependent events

❌ P(two aces without replacement) = 4/52 × 4/52
✓ P(two aces without replacement) = 4/52 × 3/51

Error 3: Confusing with/without replacement

Always check: Does the first event affect the second?

Tips for Success

Tip 1: Identify if events are independent or dependent

Tip 2: "And" usually means multiply

Tip 3: Without replacement → probabilities change

Tip 4: Draw tree diagrams for complex problems

Tip 5: Use complement for "at least one" problems

Tip 6: Check that all probabilities sum to 1

Tip 7: Simplify fractions for final answers

Summary

Independent events: P(A and B) = P(A) × P(B)

Dependent events: P(A and B) = P(A) × P(B|A)

With replacement: Independent

Without replacement: Dependent

Tree diagrams: Show all outcomes and calculate probabilities

Complement rule: P(at least one) = 1 − P(none)

Applications: Cards, dice, sampling, genetics, quality control

Practice