Unit E: Statistics and Probability

Chapter 2: Probability

Finding the Odds in Favour

In Lesson 1, probability was defined as the chance of an event happening. Probability compares the number of favourable outcomes to the number of total outcomes. The formula for probability is

probability = \frac{number of favourable outcomes}{total number of possible outcomes}

Any probability can be written in terms of odds rather than probabilities.

Odds state the chance of an event happening differently than probability. Odds in favour compare the number of favourable outcomes to the number of unfavourable outcomes. Odds are always stated as a ratio or statement. The formula used to calculate odds in favour is

odds in favour = number of favourable outcomes : number of unfavourable outcomes

It is important to understand the difference between probability and odds.

Probability	Odds in Favour
can be expressed as a fraction, decimal, percent, ratio, or statement	can only be expressed as a ratio or statement
compare favourable outcomes to the total possible outcomes	compares favourable outcomes to unfavourable outcomes
is stated as favourable outcomes : total outcomes	is stated as favourable outcomes : unfavourable outcomes
must be a value between 0 and 1 when expressed as a decimal	can be expressed using any two positive numbers

Example 1

There are 7 blue blocks and 3 red blocks placed in a bag. One block is drawn from this bag.

What is the probability that a blue block will be chosen? Express the answer as a fraction.
What are the odds in favour of randomly choosing a blue block? Express the answer as a ratio.

Solutions

\begin{array}{rcl} probability & = & \frac{number of favourable outcomes}{total number of possible outcomes} \\ = & \frac{number of blue blocks}{total number of blocks} \\ = & \frac{7}{10} \end{array}

This probability can be expressed as a fraction, decimal, percentage, ratio, or statement.

fraction	$\frac{7}{10}$
decimal	0.7
percentage	70%
ratio	7:10
statement	7 to 10

It is helpful to fill out the table when calculating odds in favour.

favourable outcomes (blue block)	7
unfavourable outcomes (not a blue block)	3

Note: The sum of the two lines in the table is equal to the total number of outcomes.

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of blue blocks : number of blocks that are not blue \\ = & 7 : 3 \end{array}

The odds in favour of choosing a blue block are 7:3.

Example 2

There are 7 pennies, 5 nickels, and 3 dimes placed in a bag. One coin is randomly chosen.

What is the probability that a nickel will be chosen? Express the answer as a ratio in lowest terms.
What are the odds of randomly choosing a nickel? Express the answer as a ratio in lowest terms.
Is the probability of choosing a nickel the same as the odds in favour of choosing a nickel? Explain.
What are the odds of randomly choosing a dime? Express the answer as a ratio in lowest terms.

Solutions

$\begin{array}{rcl} probability & = & \frac{number of favourable outcomes}{total number of possible outcomes} \\ = & \frac{number of nickels}{total number of coins} \\ = & \frac{5}{15} \\ = & \frac{1}{3} \end{array}$

Note: When probability is stated as a ratio, it can be simplified in the same way as fractions. The ratio 5:15 can be reduced by dividing each number by 5 to obtain 1:3.

The probability, expressed as a ratio, of choosing a nickel is 1:3.

Note: All ratios and fractions must be expressed in lowest terms.

favourable outcomes (coin is a nickel)	5
unfavourable outcomes (coin is not a nickel)	10

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of nickels : number of coins that are not nickels \\ = & 5 : 10 \\ = & 1 : 2 \end{array}

The odds in favour of choosing a nickel are 1:2.

No, they are not the same. The probability is the chance of the nickel being chosen from all coins. The odds are the chance of the nickel being chosen compared to the other coins.

favourable outcomes (coin is a dime)	3
unfavourable outcomes (coin is not a dime)	12

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of dimes : number of coins that are not dimes \\ = & 3 : 12 \\ = & 1 : 4 \end{array}

The odds of choosing a dime are 1:4.

One method of determining the number of favourable outcomes is to count them. Another way to calculate the number of unfavourable outcomes is to use the following formula.

unfavourable outcomes = total outcomes – favourable outcomes

Example 3

A bag of hard candy has a total of 12 candies. It includes 2 grape, 3 orange, 3 lemon, and 4 watermelon candies.

What are the odds of randomly choosing a lemon candy from the bag? Express the answer as a ratio in lowest terms.
What are the odds of randomly choosing a grape candy from the bag? Express the answer as a ratio in lowest terms.

Solutions

The number of favourable outcomes (lemon candy) is 3.

Use the formula to find the number of unfavourable outcomes (not a lemon candy).

\begin{array}{rcl} unfavourable outcomes & = & total outcomes - favourable outcomes \\ = & total outcomes - number of lemon candies \\ = & 12 - 3 \\ = & 9 \end{array}

favourable outcomes (candy is lemon)	3
unfavourable outcomes (candy is not lemon)	9

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of lemon candies : number of candies that are not lemon \\ = & 3 : 9 \\ = & 1 : 3 \end{array}

The odds of choosing a lemon candy are 1:3.

\begin{array}{rcl} unfavourable outcomes & = & total outcomes - favourable outcomes \\ = & total outcomes - number of grape candies \\ = & 12 - 2 \\ = & 10 \end{array}

favourable outcomes (candy is grape)	2
unfavourable outcomes (candy is not grape)	10

\begin{array}{rcl} odds in favour & = & number of favourable outcomes : number of unfavourable outcomes \\ = & number of grape candies : number of candies that are not grape \\ = & 2 : 10 \\ = & 1 : 5 \end{array}

The odds of choosing a grape candy are 1:5.