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Probability Representations

Introduction to Probability

Probability is a measure of how likely an event is to occur. It can be expressed in different forms:

  • As a fraction between 0 and 1
  • As a decimal between 0 and 1
  • As a percentage between 0% and 100%
  • As a ratio comparing favorable to unfavorable outcomes

The probability of an event is calculated as:

\[ P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]

Impossible Event:

Probability = 0 (or 0%)

Certain Event:

Probability = 1 (or 100%)

Probability as Fractions

Understanding Fractional Probability

When expressing probability as a fraction:

  • The numerator represents the number of favorable outcomes
  • The denominator represents the total number of possible outcomes
  • The fraction should always be in its simplest form

Example 1: What is the probability of rolling a 3 on a standard die?

\[ P(3) = \frac{1}{6} \]

Example 2: A bag contains 4 red marbles and 6 blue marbles. What is the probability of drawing a red marble?

\[ P(\text{Red}) = \frac{4}{10} = \frac{2}{5} \]

Practice Exercise

Question 1

What is the probability of getting heads when flipping a fair coin? Express as a fraction.

There are 2 possible outcomes (heads or tails), with 1 favorable outcome (heads).

\[ P(\text{Heads}) = \frac{1}{2} \]

Question 2

A spinner has 8 equal sections numbered 1-8. What is the probability of landing on an even number?

There are 4 favorable outcomes (2, 4, 6, 8) out of 8 possible outcomes.

\[ P(\text{Even}) = \frac{4}{8} = \frac{1}{2} \]

Question 3

A jar contains 5 green, 3 red, and 2 yellow candies. What is the probability of randomly selecting a red candy?

There are 3 favorable outcomes (red) out of 10 total candies.

\[ P(\text{Red}) = \frac{3}{10} \]

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Probability as Decimals

Converting Fractions to Decimals

To express probability as a decimal:

  1. First express the probability as a fraction
  2. Divide the numerator by the denominator
  3. Round if necessary (usually to 2 or 3 decimal places)

Example 1: Convert P(Heads) = 1/2 to decimal form

\[ \frac{1}{2} = 0.5 \]

Example 2: Convert P(Rolling a 1 on a die) = 1/6 to decimal form

\[ \frac{1}{6} \approx 0.1667 \]

Decimal Range:

Probability as a decimal always falls between 0 (impossible) and 1 (certain)

Practice Exercise

Question 1

Convert the probability of rolling a 5 on a standard die to decimal form.

P(5) = 1/6 ≈ 0.1667

Question 2

A bag has 3 blue and 7 red marbles. What is P(Blue) in decimal form?

P(Blue) = 3/10 = 0.3

Question 3

Convert the probability of drawing an ace from a standard deck (52 cards) to decimal form.

P(Ace) = 4/52 ≈ 0.0769

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Probability as Percentages

Converting to Percentages

To express probability as a percentage:

  1. First express the probability as a decimal
  2. Multiply by 100
  3. Add the percent sign (%)

Example 1: Convert P(Heads) = 0.5 to percentage

\[ 0.5 \times 100 = 50\% \]

Example 2: Convert P(Rolling a 1) ≈ 0.1667 to percentage

\[ 0.1667 \times 100 \approx 16.67\% \]

Percentage Range:

Probability as a percentage always falls between 0% (impossible) and 100% (certain)

Practice Exercise

Question 1

Convert the probability of rolling an even number on a die to percentage.

P(Even) = 3/6 = 0.5 = 50%

Question 2

A weather forecast says there's a 0.35 probability of rain. Express this as a percentage.

0.35 × 100 = 35%

Question 3

Convert P(Drawing a heart from a standard deck) to percentage.

P(Heart) = 13/52 = 0.25 = 25%

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Probability as Ratios

Understanding Probability Ratios

Probability can be expressed as a ratio comparing:

  • Favorable outcomes to unfavorable outcomes (a:b)
  • Or favorable outcomes to total outcomes (a out of b)

Example 1: Express P(Heads) as a ratio

Favorable:Unfavorable = 1:1

Or 1 out of 2

Example 2: A bag has 3 red and 5 blue marbles. Express P(Red) as a ratio.

Favorable:Unfavorable = 3:5

Or 3 out of 8

Practice Exercise

Question 1

Express the probability of rolling a 2 on a die as a ratio of favorable to unfavorable outcomes.

Favorable:Unfavorable = 1:5

Question 2

A class has 12 boys and 18 girls. Express P(Boy) as a ratio.

Favorable:Unfavorable = 12:18 = 2:3

Or 12 out of 30

Question 3

A lottery has 1 winning ticket out of 100. Express the probability of winning as a ratio.

Favorable:Unfavorable = 1:99

Or 1 out of 100

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Using Graphic Organizers

Probability Tables

Probability tables help organize possible outcomes and their probabilities:

Example: Probability distribution for rolling two dice

Probability table for two dice

Tree Diagrams

Tree diagrams show all possible outcomes of sequential events:

Example: Possible outcomes when flipping a coin twice

Probability tree diagram

Other Graphic Organizers

Other useful tools for representing probability include:

  • Venn diagrams
  • Bar charts
  • Pie charts
  • Two-way tables

Practice Exercise

Question 1

Create a probability table for spinning a spinner with equal red, blue, and green sections.

Spinner probability table

Question 2

Draw a tree diagram showing possible outcomes when rolling a die and then flipping a coin.

Die and coin tree diagram

Question 3

A bag contains 4 red and 6 blue marbles. Create a graphic organizer showing probabilities for drawing two marbles with replacement.

Marble probability tree

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Math Challenge

Timed Test in 7 Levels

Test your probability skills through progressively challenging levels

Each level has 20 questions. Need 17 correct to advance.