Mutually Exclusive Events - Definition, Formula, Examples - Free Printable

Let’s go through each problem one by one. We’ll figure out if the events are mutually exclusive (ME) or non-mutually exclusive (non), and then calculate the probability.

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1. Choosing a penny or a dime from 4 pennies, 3 nickels, and 6 dimes.

Total coins = 4 + 3 + 6 = 13

- Penny: 4
- Dime: 6

Can you pick a coin that is both a penny AND a dime? → No → So these are mutually exclusive (ME).

Probability = P(penny) + P(dime) = 4/13 + 6/13 = 10/13

✔ ME
✔ Probability = 10/13

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2. Use the table:

| | Blond-Haired | Not Blond-Haired | Total |
|----------|--------------|------------------|-------|
| Boys | 6 | 9 | 15 |
| Girls | 5 | 7 | 12 |
| Total| 11 | 16 | 27|

#### a. Probability of calling on a blond-haired person OR a boy?

Blond-haired total = 11
Boys total = 15
But — some boys are blond-haired! (6 of them). So if we just add 11 + 15, we’re double-counting those 6.

→ So this is non-mutually exclusive (non)

Use formula:
P(A or B) = P(A) + P(B) - P(A and B)

= 11/27 + 15/27 - 6/27 = (11 + 15 - 6)/27 = 20/27

✔ non
✔ Probability = 20/27

#### b. Probability of calling on a blond-haired girl OR a boy?

Blond-haired girls = 5
Boys = 15

Is there any overlap? Can someone be both a blond-haired girl AND a boy? → No → Mutually exclusive!

So: P = 5/27 + 15/27 = 20/27

Wait — same number as part a? Coincidence! But yes, because in part a we subtracted the overlap, here there’s no overlap to subtract.

✔ ME
✔ Probability = 20/27

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3. Drawing a diamond OR queen from a standard deck (52 cards)

Diamonds: 13 cards
Queens: 4 cards (one per suit)
BUT — one card is BOTH a diamond AND a queen → Queen of Diamonds

So they overlap → non-mutually exclusive

P = P(diamond) + P(queen) - P(both)
= 13/52 + 4/52 - 1/52 = 16/52 = 4/13

✔ non
✔ Probability = 4/13

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4. Tossing two dice and showing at least one 4.

This means: first die is 4, OR second die is 4, OR both.

Are these mutually exclusive? No — because you can get (4,4), which counts for both.

Actually, “at least one 4” includes:
- First die 4, second not 4 → 5 outcomes: (4,1),(4,2),(4,3),(4,5),(4,6)
- Second die 4, first not 4 → 5 outcomes: (1,4),(2,4),(3,4),(5,4),(6,4)
- Both 4 → 1 outcome: (4,4)

Total favorable = 5 + 5 + 1 = 11

Total possible outcomes when rolling two dice = 6 × 6 = 36

So probability = 11/36

And since (4,4) is counted in both “first is 4” and “second is 4”, it’s non-ME

✔ non
✔ Probability = 11/36

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5. Selecting an ace OR a red card from a deck.

Aces: 4 total (2 red, 2 black)
Red cards: 26 total (hearts and diamonds)

Overlap: 2 red aces → so not mutually exclusive → non

P = P(ace) + P(red) - P(red ace)
= 4/52 + 26/52 - 2/52 = 28/52 = 7/13

✔ non
✔ Probability = 7/13

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6. Two dice tossed, sum is 6 OR sum is 9.

List sums:

Sum = 6: (1,5), (2,4), (3,3), (4,2), (5,1) → 5 ways
Sum = 9: (3,6), (4,5), (5,4), (6,3) → 4 ways

Any overlap? Is there a roll that gives both sum 6 and sum 9? → No → Mutually exclusive!

Total favorable = 5 + 4 = 9
Total outcomes = 36

Probability = 9/36 = 1/4

✔ ME
✔ Probability = 1/4

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7. Bag has cards numbered 1 to 14. Draw one at random.

Sample space: {1,2,3,4,5,6,7,8,9,10,11,12,13,14} → 14 cards

#### a) Prime number OR multiple of four

Primes between 1–14: 2,3,5,7,11,13 → 6 numbers
Multiples of 4: 4,8,12 → 3 numbers

Overlap? Any prime that’s also multiple of 4? → No → ME

Total favorable = 6 + 3 = 9
Probability = 9/14

✔ ME
✔ Probability = 9/14

#### b) Multiple of two OR multiple of three

Multiples of 2: 2,4,6,8,10,12,14 → 7 numbers
Multiples of 3: 3,6,9,12 → 4 numbers

Overlap: numbers that are multiples of BOTH 2 and 3 → multiples of 6: 6,12 → 2 numbers

So non-ME

P = 7/14 + 4/14 - 2/14 = 9/14

✔ non
✔ Probability = 9/14

#### c) Selecting a 3 OR a 4

Only one card is 3, only one is 4 → can’t be both → ME

P = 1/14 + 1/14 = 2/14 = 1/7

✔ ME
✔ Probability = 1/7

#### d) Selecting an 8 OR a number less than 8

Numbers less than 8: 1,2,3,4,5,6,7 → 7 numbers
Plus 8 → but 8 is NOT less than 8 → so no overlap → ME

Favorable: 7 (less than 8) + 1 (the 8) = 8 numbers

P = 8/14 = 4/7

✔ ME
✔ Probability = 4/7

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Final Answer:

1. ME; 10/13
2a. non; 20/27
2b. ME; 20/27
3. non; 4/13
4. non; 11/36
5. non; 7/13
6. ME; 1/4
7a. ME; 9/14
7b. non; 9/14
7c. ME; 1/7
7d. ME; 4/7