Probability of Independent & Mutually Exclusive Events

The basic idea: favourable over total

Probability is always a fraction (which you can also write as a decimal or a percent). The recipe never changes:

• Favourable means the outcomes you are hoping for, the "wins."
• Total means every outcome that could possibly happen, counted fairly.
• The answer is always between 0 and 1. A probability of 0 means it can never happen; 1 means it is certain. Nothing is ever negative or above 1.

Worked example #1: one roll of a die

A standard die has faces 1, 2, 3, 4, 5, 6. What is the probability of rolling an even number?

• Favourable outcomes: the even faces are 2, 4, and 6, that is 3 of them.
• Total outcomes: there are 6 faces in all.
• Simplify: 3⁄6 reduces to 1⁄2.

P(even) = favourable ÷ total
        = 3 ÷ 6
        = 1/2   (= 0.5 = 50%)

Independent events: the "and" rule (multiply)

Two events are independent when the first one does not change the odds of the second. Flipping a coin does not affect the next coin; rolling a die today does not affect tomorrow's roll. When you want both independent events to happen, you multiply their probabilities. This formula is on your sheet:

P(A and B) = P(A) × P(B)

Worked example #2: a coin and a die together

You flip a fair coin and roll a die. What is the probability of getting heads and a 5? The coin and the die are independent, so we multiply.

• P(heads): 1 winning face out of 2, so 1⁄2.
• P(rolling a 5): 1 winning face out of 6, so 1⁄6.
• Multiply (the "and" rule): 1⁄2 × 1⁄6. Multiply the tops, multiply the bottoms.

P(heads and 5) = P(heads) × P(5)
               = 1/2 × 1/6
               = (1 × 1) / (2 × 6)
               = 1/12   (≈ 0.083 ≈ 8.3%)

Mutually exclusive events: the "or" rule (add)

Two events are mutually exclusive when they cannot both happen at the same time. On a single die you cannot roll a 2 and a 5 on the same roll, it is one or the other. When you want either of two mutually exclusive events, you add their probabilities. This formula is also on your sheet:

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

A picture: a spinner with separate sections

This spinner is split into 8 equal sections: 3 red, 2 blue, 2 green, and 1 yellow. Each spin lands on exactly one colour, so the colours are mutually exclusive, you can never get red and blue on a single spin. That is why we can add to find "red or blue."

8 equal sections in all. P(red or blue) = 3⁄8 + 2⁄8 = 5⁄8.

Worked example #3: rolling a 2 or a 5

You roll one die. What is the probability of getting a 2 or a 5? On a single roll you cannot land on both, so these events are mutually exclusive, we add.

• P(roll a 2): 1 out of 6, so 1⁄6.
• P(roll a 5): 1 out of 6, so 1⁄6.
• Add (the "or" rule): with the same bottom number, just add the tops.

P(2 or 5) = P(2) + P(5)
          = 1/6 + 1/6
          = 2/6
          = 1/3   (≈ 0.333 ≈ 33%)

Worked example #4: drawing a king or a queen

A standard deck has 52 cards. It holds 4 kings and 4 queens. You draw one card. What is the probability it is a king or a queen? A single card cannot be both a king and a queen, so these are mutually exclusive, add them.

• P(king): 4 kings out of 52, so 4⁄52.
• P(queen): 4 queens out of 52, so 4⁄52.
• Add, then simplify: 4⁄52 + 4⁄52 = 8⁄52, which reduces to 2⁄13.

P(king or queen) = P(king) + P(queen)
                 = 4/52 + 4/52
                 = 8/52
                 = 2/13   (≈ 0.154 ≈ 15%)

"And" vs "or" at a glance

The whole lesson comes down to spotting one word. Here is the side by side:

P(A and B) = P(A) × P(B)

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

Both formulas are printed on the CAEC formula sheet, so you do not have to memorize them. Your job is to read the question, find the and or the or, and pick the matching rule.

Tips that keep you from slipping

• Circle the keyword. Underline "and" or "or" in the question. "And" means multiply; "or" means add. That one habit prevents most mistakes.
• Count total outcomes carefully. A die has 6 faces, a coin has 2, a deck has 52. Get the bottom number right before anything else.
• Sanity-check the size. An "and" answer should be smaller than either piece; an "or" answer should be larger. If yours went the wrong way, you used the wrong rule.
• Use your calculator to convert. This is the calculator section, so divide top by bottom to turn any fraction into a decimal or percent if the answer choices are in that form.

Your turn: practice problems

Find the keyword, pick multiply or add, then solve. No peeking until you have tried.

1. You flip a fair coin twice. What is the probability of heads and heads (heads both times)?
2. You roll one die. What is the probability of rolling a 1 or a 6?
3. A spinner has 8 equal sections: 3 red, 2 blue, 2 green, 1 yellow. What is the probability of landing on red or green?
4. You roll a die and flip a coin. What is the probability of an even number and tails?