Probability Basics

For Elementary Students

What Is Probability?

Probability tells you how likely something is to happen.

Think about it like this: Will it rain today? Will you roll a 6? Will you pick a red marble? Probability helps you answer these questions!

Probability Words

We use special words to describe how likely things are:

• Impossible — will never happen (0% chance)
• Unlikely — probably won't happen
• Equally likely — might or might not happen (50-50 chance)
• Likely — probably will happen
• Certain — will definitely happen (100% chance)

Examples of Probability

Impossible (0%):

• Rolling a 7 on a regular die (only goes up to 6!)
• Picking a purple marble from a bag of only red and blue marbles

Certain (100%):

• The sun will rise tomorrow
• Picking a red marble from a bag of all red marbles

Equally Likely (50%):

• Flipping a coin and getting heads
• Picking a red or blue marble when there are equal amounts

Flipping a Coin

Question: What's the probability of getting heads?

Think about it:

• How many ways can the coin land? 2 ways (heads or tails)
• How many of those are heads? 1 way

Answer: 1 out of 2, or 1/2

Rolling a Die

Question: What's the probability of rolling a 4?

A die has 6 sides: 1, 2, 3, 4, 5, 6

• Total possibilities: 6
• Ways to get a 4: 1

Answer: 1 out of 6, or 1/6

Picking from a Bag

Question: A bag has 2 red marbles and 3 blue marbles. What's the probability of picking red?

🔴 🔴 🔵 🔵 🔵

• Total marbles: 2 + 3 = 5
• Red marbles: 2

Answer: 2 out of 5, or 2/5

For Junior High Students

Understanding Probability

Probability measures the likelihood of an event occurring.

Range: Probability is always a number between 0 and 1:

• 0 = impossible
• 1 = certain
• 0.5 = equally likely (50-50)

Also written as:

• Fractions (1/2)
• Decimals (0.5)
• Percentages (50%)

The Probability Formula

For equally likely outcomes:

P(event) = (number of favorable outcomes) / (total number of outcomes)

Or more simply:

P(event) = favorable / total

Where:

• P(event) means "probability of the event"
• Favorable = outcomes you want
• Total = all possible outcomes

Example 1: Coin Flip

Question: Probability of getting heads?

• Favorable: 1 (heads)
• Total: 2 (heads or tails)
• P(heads) = 1/2 = 0.5 = 50%

Example 2: Rolling a Die

Question: Probability of rolling a 3?

• Favorable: 1 (the number 3)
• Total: 6 (numbers 1, 2, 3, 4, 5, 6)
• P(3) = 1/6 ≈ 0.167 ≈ 16.7%

Question: Probability of rolling an even number?

• Favorable: 3 (the numbers 2, 4, 6)
• Total: 6
• P(even) = 3/6 = 1/2 = 50%

Example 3: Drawing from a Bag

Question: A bag has 3 red balls and 7 blue balls. Probability of picking red?

• Favorable: 3 (red balls)
• Total: 3 + 7 = 10 (all balls)
• P(red) = 3/10 = 0.3 = 30%

Expressing Probability Three Ways

All three forms mean the same thing!

Complementary Events

The probability of something NOT happening is:

P(not A) = 1 − P(A)

Example: If the probability of rain is 30% (0.3), what's the probability of NO rain?

• P(no rain) = 1 − 0.3 = 0.7 = 70%

Why? All probabilities must add up to 100% (or 1).

Certain, Impossible, and In-Between

Impossible: P = 0

• Rolling an 8 on a 6-sided die

Unlikely: 0 < P < 0.5

• Rolling a 6 on a die: P = 1/6 ≈ 0.17

Equally likely: P = 0.5

• Coin landing on heads

Likely: 0.5 < P < 1

• NOT rolling a 6: P = 5/6 ≈ 0.83

Certain: P = 1

• Rolling a number between 1 and 6 on a die

Sample Space

Sample space = all possible outcomes

Example: Flipping a coin

• Sample space: {Heads, Tails}
• Size: 2

Example: Rolling a die

• Sample space: {1, 2, 3, 4, 5, 6}
• Size: 6

Probability with Spinners

Question: A spinner has 8 equal sections: 3 red, 2 blue, 3 green. Probability of landing on blue?

• Favorable: 2 (blue sections)
• Total: 8 (all sections)
• P(blue) = 2/8 = 1/4 = 25%

Checking Your Answer

Probability must be between 0 and 1!

If you get a negative number or something greater than 1, you made a mistake!

Example:

• ✓ P = 0.6 (valid)
• ✓ P = 1/3 (valid)
• ❌ P = 1.5 (impossible! Too high)
• ❌ P = −0.2 (impossible! Negative)

Real-Life Probability

Weather: "30% chance of rain" means P(rain) = 0.3

Games: "1 in 6 chance of winning" means P(win) = 1/6

Sports: "50-50 game" means each team has P = 0.5 to win

Lotteries: Very low probability (like 1 in a million)

Important Notes

Fair outcomes: This formula only works when all outcomes are equally likely!

• Fair coin: heads and tails equally likely ✓
• Weighted die: sides NOT equally likely ❌

Random: Results must be random (unpredictable)

Practice