Lesson 11
Introduction to Probability and Counting Outcomes
Learning Goals
To identify the properties of probability and evaluate the probability of single events.
To describe and evaluate the probability of multiple events.
To use counting techniques (tree diagrams, permutations, combinations) to evaluate probability.
To solve probability problems that occur in our everyday lives.
Important Terms
Trial ­ one repetition of an experiment
eg. Rolling two dice one time.
Outcome or Event ­ a given result within an experiment
eg. Rolling a sum of seven with two dice.
Sample Space ­ the set of all possible outcomes
eg. All the possible sums that could result when rolling two dice.
Introduction to Probability
Probability is the measured likelihood of a given event occurring.
The likelihood of a given event occurring will vary from completely impossible to absolutely certain.
Impossible
Certain
Probability is measured using fractions and percentages and can be expressed as a fraction, percent or a ratio.
Examples
Jen makes it over the high jump 2 out of every 5 jumps.
There is a 60% chance of rain.
The odds of a horse winning a race is 3:1.
Experimental Probability
For a given event A, P(A) = number of times event A occurred
number of trials
Minds On ­ Try It! What is the experimental probability of flipping two heads if you flip two coins a total of 20 times?
Experiment: You flip 2 coins.
Trial ­ each time you flip 2 coins
Outcomes ­ There are 3 possible outcomes:
2 heads {H, H}, 2 tails {T, T}, and 1 head, 1 tail {H, T}
Flip the coins a total of 20 times and record how many times you get two heads.
The experimental probability will be determined by the formula:
Exp. Probability = number of times you achieve the outcome
of interest Total number of trials
Try this now. Determine the outcome of interest and find the experimental probability of that event occurring.
Theoretical Probability
Theoretical probability is the ‘true’ probability of a given event occurring, which can be predicted mathematically (using permutation and combination theory)
The theoretical probability must be a value between 0 and 1.
The theoretical probability of an event A occurring is determined by:
P(A) = number of ways to achieve Event A
total number of possible outcomes
or P(A) = n(A)
n(S)
where n(A) is the total number of outcomes that represent event A and n(S) is the total number of possible outcomes.
This method of determining a probability is based on the assumption that all outcomes are equally likely.
To determine the total number of outcomes for n(A) and n(S) we must use the theories of counting principles. Strategies of Counting Principles
1) Tree diagrams and outcome tables
2) Permutation Theory
3) Combination Theory
Lets return to our example from before to examine how we use counting principles to find theoretical probabilities.
Minds On ­ Try It! What is the theoretical probability of flipping two heads if you flip two coins a total of 20 times?
Solution
Begin by drawing a tree diagram to determine all of the possible outcomes when flipping two coins.
1st Coin
2nd Coin
H
H
T
T
H
T
So out of 20 tries we would expect to see:
Read Examples 1 ­3 on pages 8­11.
Read Examples 1 ­ 3 on pages 18­22.
Practice
Section 1.1, pages 13­15, #1­3, 6, 7­9
Section 1.2, pages 24­25, #1­6
The Probability of Multiple Events
To evaluate the probability of multiple events, we must consider whether the events of interest occur together, at the same time, or separately, as this or that.
If the events of interest occur together, at the same time, then we use the Rule of Product.
We find the probability of event A and multiply by the probability of event B, and event C, and so on.
Example
What is the probability that you roll a five with a single die and then draw an ace from a deck of cards?
Let event A be rolling a five with a single die.
Let event B be drawing an ace from a deck of cards.
We will do these two actions together, therefore:
P(A) = 1 P(B) = 1
6 13
P(A and B) = 1 x 1
6 13
= 1 78
The Probability of Multiple Events
To evaluate the probability of multiple events, we must consider whether the events of interest occur together, at the same time, or separately, as this or that.
If the events of interest occur separately, or one after the other, we use the Rule of Sum.
We find the probability of event A and add it to the probability of event B, and event C, and so on.
Example
What is the probability of rolling a three or a five with a single die?
Let event A be rolling a three with a single die.
Let event B be rolling a five with a single die.
We will do these two actions separately, therefore:
P(A) = 1 P(B) = 1
6 6
P(A or B) = 1 + 1
6 6
= 2
6
= 1
3
Finding the Probability of an Event Not Occurring
Consider this problem:
A single die is rolled. What is the probability that a two is not rolled?
We can evaluate this problem in two ways.
Method 1 ­ Identify the number of ways to not roll a two.
1, 3, 4, 5, 6
Therefore n(A) = 5, n(S) = 6, so P(A) = 5/6
Method 2 ­ Identify the number of ways to roll a two.
n(A) = 1 n(S) = 6, so P(A) = 1/6
The probability of rolling any number is 1.
Therefore the probability of not rolling a two is:
1 ­ 1
6
= 6 ­ 1
6
= 5
6
Therefore the probability of an event A not occurring can be determined by:
P(A') = 1 ­ P(A)
Using Counting Strategies to Determine Probabilities
In the following examples we will look at how we use the counting strategies we have studied to determine probabilities.
Example 1
What is the probability of rolling a sum greater than 6 when rolling two dice?
Solution ­ Use an Outcome Table or Tree Diagram Strategy
2nd Die
1
2
3
4
5
6
1
1st Die
2
3
4
5
6
Example 2 A lottery is made up of 5 numbers, using the digits 0 ­ 9. What is the probability that you choose the winning numbers?
Solution­ Using Permutation Strategies
Example 3
A fundraiser is held by a youth group. All participants in the fundraiser are entered into a draw to win a trip to Toronto. There are 6 girls and 4 boys who participated in the fundraiser and there can be four winners. What is the probability that the winners are made up of:
a) all girls
b) all boys
c) 2 girls and 2 boys
Solution ­ Using Combination Strategies
Practice
Section 2.5, pages 93­95, #1­6, 8­10
Section 3.5, pages 132­133, #1­9, 13, 15