🎲 Probability - Understanding Chance
What is Probability?
Probability is the chance that something will happen!
It tells us how likely an event is to occur - from impossible to certain.
Probability = The likelihood of something happening
📊 The Probability Scale
Events can be placed on a scale from Impossible to Certain:
\(0\%\)
\(25\%\)
\(50\%\)
\(75\%\)
\(100\%\)
📚 Probability Vocabulary
❌ Impossible
Impossible means an event will NEVER happen. There is \(0\%\) chance!
Formula:
Probability of Impossible Event \(= 0\) or \(0\%\)
Examples:
- 🦄 Riding a unicorn to school
- 🌙 Flying to the moon without a spaceship
- 🐕 Your dog speaking English
- 🎲 Rolling a \(7\) on a standard die (die only has \(1-6\))
- 🔢 Picking a purple marble from a bag with only red and blue marbles
⚠️ Unlikely (or Improbable)
Unlikely means an event probably won't happen, but it could!
The chance is low - less than \(50\%\).
Formula:
\(0 < \text{Probability} < 0.5\) or \(0\% < \text{Probability} < 50\%\)
Examples:
- 🌨️ It will snow in summer
- 🎯 Hitting a bullseye on your first try
- 🎲 Rolling a \(6\) on a die (only \(1\) out of \(6\) chances)
- 🔵 Picking a blue marble from a bag with \(2\) blue and \(8\) red marbles
- 🏆 Winning the lottery
⚖️ Equally Likely (Equal Probability)
Equally likely means two or more events have the same chance of happening!
Each event has exactly \(50\%\) chance (for two events).
Formula:
Probability of Event A \(=\) Probability of Event B
\(P(A) = P(B) = 0.5\) or \(50\%\) (for two events)
Examples:
- 🪙 Flipping a coin: heads or tails (both \(50\%\))
- 🔴🔵 Bag with \(5\) red and \(5\) blue marbles (equal chance)
- 🎲 Rolling an even or odd number on a die (both \(50\%\))
- 🧃 Choosing chocolate or vanilla when there are \(3\) of each
- ⬆️⬇️ Walking up or down when there are equal numbers of stairs
👍 Likely (or Probable)
Likely means an event will probably happen!
The chance is high - more than \(50\%\).
Formula:
\(0.5 < \text{Probability} < 1\) or \(50\% < \text{Probability} < 100\%\)
Examples:
- ☀️ The sun will rise tomorrow
- 🍽️ You will eat dinner tonight
- 😴 You will sleep at night
- 🔴 Picking a red marble from a bag with \(8\) red and \(2\) blue marbles
- 📚 Going to school on Monday
✅ Certain (Definite)
Certain means an event will ALWAYS happen. It's \(100\%\) sure!
Formula:
Probability of Certain Event \(= 1\) or \(100\%\)
Examples:
- 🌍 The Earth will rotate every day
- 🎂 You will get older every year
- 🔢 After \(5\) comes \(6\) when counting
- 🔴 Picking a red marble from a bag with ONLY red marbles
- 🎲 Rolling a number between \(1\) and \(6\) on a standard die
⚖️ Comparing Probabilities
How to Compare Events
We can compare the probability of different events to see which is:
- 📈 More likely - has a greater chance of happening
- 📉 Less likely - has a smaller chance of happening
- ⚖️ Equally likely - has the same chance of happening
📈 More Likely
Event A is more likely than Event B when:
\(P(A) > P(B)\)
(Probability of A is greater than Probability of B)
Example:
A bag has \(8\) red marbles and \(2\) blue marbles.
Question: Which is more likely - picking red or blue?
Answer: Picking red is more likely!
Why? Because there are more red marbles (\(8\)) than blue marbles (\(2\)).
Red: \(\frac{8}{10} = 0.8 = 80\%\)
Blue: \(\frac{2}{10} = 0.2 = 20\%\)
Since \(80\% > 20\%\), red is more likely! ✓
📉 Less Likely
Event A is less likely than Event B when:
\(P(A) < P(B)\)
(Probability of A is less than Probability of B)
Example:
A bag has \(3\) vanilla cupcakes and \(7\) chocolate cupcakes.
Question: Which is less likely - picking vanilla or chocolate?
Answer: Picking vanilla is less likely!
Why? Because there are fewer vanilla cupcakes (\(3\)) than chocolate (\(7\)).
Vanilla: \(\frac{3}{10} = 0.3 = 30\%\)
Chocolate: \(\frac{7}{10} = 0.7 = 70\%\)
Since \(30\% < 70\%\), vanilla is less likely! ✓
⚖️ Equally Likely
Event A and Event B are equally likely when:
\(P(A) = P(B)\)
(Probability of A equals Probability of B)
Example:
A bag has \(5\) green apples and \(5\) red apples.
Question: Is picking green or red more likely?
Answer: They are equally likely!
Why? Because there are the same number of each (\(5\) and \(5\)).
Green: \(\frac{5}{10} = 0.5 = 50\%\)
Red: \(\frac{5}{10} = 0.5 = 50\%\)
Since \(50\% = 50\%\), they are equally likely! ✓
📐 Basic Probability Formula
Simple Probability Formula
To find the probability of an event happening, we use this formula:
\(\text{Probability} = \frac{\text{Number of ways it can happen}}{\text{Total number of possible outcomes}}\)
OR
\(P(\text{Event}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}}\)
Step-by-Step Guide
- Count the favorable outcomes (how many ways your event can happen)
- Count all possible outcomes (total number of things that could happen)
- Divide favorable by total to get the probability
- Write as a fraction or percentage
📖 Complete Worked Examples
Example 1: Spinning a Spinner
Problem:
A spinner has \(4\) equal sections: red, blue, green, and yellow.
What is the probability of landing on blue?
Is it more likely to land on blue or on a color that is NOT blue?
Solution:
Step 1: Count favorable outcomes = \(1\) (only blue)
Step 2: Count total outcomes = \(4\) (red, blue, green, yellow)
Step 3: Calculate probability:
\(P(\text{blue}) = \frac{1}{4} = 0.25 = 25\%\)
Comparison:
• Probability of blue = \(\frac{1}{4} = 25\%\)
• Probability of NOT blue = \(\frac{3}{4} = 75\%\)
Answer: Landing on a color that is NOT blue is more likely! ✓
Example 2: Picking from a Bag
Problem:
A bag contains \(6\) red balls and \(4\) blue balls.
Which color is more likely to be picked?
Which color is less likely to be picked?
Find the probability of each color.
Solution:
Total balls: \(6 + 4 = 10\)
Probability of red:
\(P(\text{red}) = \frac{6}{10} = \frac{3}{5} = 0.6 = 60\%\)
Probability of blue:
\(P(\text{blue}) = \frac{4}{10} = \frac{2}{5} = 0.4 = 40\%\)
Comparison:
Since \(60\% > 40\%\):
• Red is MORE likely (higher probability) ✓
• Blue is LESS likely (lower probability) ✓
Example 3: Rolling a Die
Problem:
You roll a standard die (numbers \(1\) to \(6\)).
a) What is the probability of rolling a \(3\)?
b) What is the probability of rolling an even number?
c) Compare: Is rolling an even number more likely than rolling a \(3\)?
Solution:
a) Probability of rolling a 3:
Favorable outcomes: \(1\) (only the number \(3\))
Total outcomes: \(6\) (numbers \(1, 2, 3, 4, 5, 6\))
\(P(3) = \frac{1}{6} \approx 16.7\%\)
b) Probability of rolling an even number:
Even numbers: \(2, 4, 6\)
Favorable outcomes: \(3\)
Total outcomes: \(6\)
\(P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5 = 50\%\)
c) Comparison:
\(P(\text{even}) = 50\%\) and \(P(3) = 16.7\%\)
Since \(50\% > 16.7\%\):
Answer: Rolling an even number is MORE likely than rolling a \(3\)! ✓
📊 Quick Reference Chart
Term | Probability Range | Meaning | Example |
---|---|---|---|
Impossible | \(0\) or \(0\%\) | Will never happen | Rolling a \(7\) on a die |
Unlikely | \(0 < P < 0.5\) | Probably won't happen | Snow in summer |
Equally Likely | \(0.5\) or \(50\%\) | Same chance | Flipping a coin |
Likely | \(0.5 < P < 1\) | Probably will happen | The sun will rise |
Certain | \(1\) or \(100\%\) | Will always happen | Earth will rotate |
📝 Important Formulas Summary
Main Probability Formula:
\(P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}\)
Probability Range:
\(0 \leq P(\text{Event}) \leq 1\)
OR
\(0\% \leq P(\text{Event}) \leq 100\%\)
Comparison Formulas:
More Likely: \(P(A) > P(B)\)
Less Likely: \(P(A) < P(B)\)
Equally Likely: \(P(A) = P(B)\)
Total Probability Rule:
\(P(\text{Event happens}) + P(\text{Event doesn't happen}) = 1\)
💡 Tips for Success
- ✓ Look at the numbers: More of something = more likely to pick it!
- ✓ Count carefully: Count favorable outcomes AND total outcomes
- ✓ Think about real life: Has it happened before? Will it happen?
- ✓ Use probability words: Impossible, unlikely, equally likely, likely, certain
- ✓ Compare fractions: Bigger fraction = more likely!
- ✓ Remember the scale: \(0\) (impossible) to \(1\) (certain)
- ✓ Equal amounts: Same number of each = equally likely
- ✓ Practice with examples: Use coins, dice, marbles, or cards!
- ✓ Check your answer: Does it make sense? Is it reasonable?