Introduction to Probability
Probability is a measure of how likely an event is to occur. It can be expressed in different ways: as fractions, decimals, percentages, or ratios. In this lesson, we'll learn how to represent probabilities in all these forms.
The probability of an event ranges from 0 (impossible) to 1 (certain). We can calculate probability using the formula:
Key Concepts:
- Probability of 0 means the event will never happen
- Probability of 1 means the event will certainly happen
- Probabilities between 0 and 1 indicate varying likelihoods
Probability as Fractions
Fractions are the most natural way to represent probability because the probability formula itself is a fraction (favorable outcomes over total outcomes).
Example 1: What is the probability of rolling a 3 on a standard die?
Number of favorable outcomes (rolling a 3) = 1
Total possible outcomes = 6 (numbers 1 through 6)
\[ P(3) = \frac{1}{6} \]Example 2: A bag contains 4 red marbles and 6 blue marbles. What is the probability of drawing a red marble?
Favorable outcomes (red marbles) = 4
Total marbles = 4 + 6 = 10
\[ P(\text{Red}) = \frac{4}{10} = \frac{2}{5} \]Practice Exercise
Question 1
A spinner has 8 equal sections numbered 1-8. What is the probability of landing on an even number as a fraction?
Even numbers: 2, 4, 6, 8 (4 outcomes)
Total sections: 8
\[ P(\text{Even}) = \frac{4}{8} = \frac{1}{2} \]Question 2
A deck of cards has 52 cards with 13 hearts. What is the probability of drawing a heart as a fraction?
Number of hearts: 13
Total cards: 52
\[ P(\text{Heart}) = \frac{13}{52} = \frac{1}{4} \]Probability as Decimals
Decimal probabilities are simply fraction probabilities converted to decimal form by dividing the numerator by the denominator.
Example 1: Convert the probability of rolling a 3 (\(\frac{1}{6}\)) to decimal form.
\[ \frac{1}{6} \approx 0.1667 \]Example 2: A weather forecast says there's a \(\frac{3}{10}\) chance of rain. Express this as a decimal.
\[ \frac{3}{10} = 0.3 \]Remember:
- Decimal probabilities range from 0.0 to 1.0
- 0.5 represents a 50% chance (equal probability)
- You can convert fractions to decimals by division
Practice Exercise
Question 1
Convert the probability \(\frac{3}{4}\) to decimal form.
Question 2
If the probability of an event is \(\frac{2}{5}\), what is its decimal equivalent?
Probability as Percentages
Percentages are another common way to express probability. To convert a decimal probability to a percentage, multiply by 100.
Example 1: Convert the probability 0.25 to a percentage.
\[ 0.25 \times 100 = 25\% \]Example 2: A survey shows \(\frac{7}{20}\) of students walk to school. Express this as a percentage.
First convert to decimal: \(\frac{7}{20} = 0.35\)
Then to percentage: \(0.35 \times 100 = 35\%\)
Key Points:
- 0% means impossible
- 100% means certain
- 50% means equally likely to happen or not happen
Practice Exercise
Question 1
Express the probability \(\frac{3}{8}\) as a percentage.
First convert to decimal: \(\frac{3}{8} = 0.375\)
Then to percentage: \(0.375 \times 100 = 37.5\%\)
Question 2
Convert the decimal probability 0.08 to a percentage.
Probability as Ratios
Probability can also be expressed as a ratio of favorable outcomes to unfavorable outcomes.
Example 1: The probability of rolling a 2 on a die is \(\frac{1}{6}\). Express this as a ratio.
Favorable outcomes: 1 (rolling a 2)
Unfavorable outcomes: 5 (rolling 1,3,4,5,6)
Ratio = 1:5
Example 2: A bag has 3 red marbles and 7 blue marbles. What is the ratio probability of drawing a red marble?
Favorable: 3 red
Unfavorable: 7 blue
Ratio = 3:7
Practice Exercise
Question 1
A class has 12 boys and 18 girls. Express the probability of selecting a boy as a ratio.
Favorable: 12 boys
Unfavorable: 18 girls
Ratio = 12:18 = 2:3 (simplified)
Question 2
In a deck of cards, what is the ratio of drawing a face card (12 face cards in 52-card deck)?
Favorable: 12 face cards
Unfavorable: 40 non-face cards (52-12)
Ratio = 12:40 = 3:10 (simplified)
Using Tree Diagrams
Tree diagrams help visualize probabilities, especially for compound events (events with multiple stages).
Example: A coin is flipped twice. Create a tree diagram and find the probability of getting one head and one tail (in any order).
From the tree diagram, there are 4 possible outcomes:
- Head then Head (HH)
- Head then Tail (HT)
- Tail then Head (TH)
- Tail then Tail (TT)
Favorable outcomes for one head and one tail: HT and TH (2 outcomes)
Total outcomes: 4
\[ P(\text{One head and one tail}) = \frac{2}{4} = \frac{1}{2} \]This can be expressed as:
- Fraction: \(\frac{1}{2}\)
- Decimal: 0.5
- Percentage: 50%
- Ratio: 1:1
Practice Exercise
Question 1
A spinner has equal red, blue and green sections. It's spun twice. Draw a tree diagram and find the probability of getting the same color both times, expressed as a fraction, decimal, percentage, and ratio.
Possible same-color outcomes: RR, BB, GG (3 outcomes)
Total outcomes: 9 (3 choices first spin × 3 choices second spin)
Fraction: \(\frac{3}{9} = \frac{1}{3}\)
Decimal: ≈ 0.333
Percentage: ≈ 33.3%
Ratio: 1:2 (1 favorable to 2 unfavorable)
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