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Honors: 9. Probability
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Grade 912

Honors: 9.01 Subsets and sample spaces

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Concept summary

To find the probability of an event we need to find the sample space for an event and the required subsets of the sample space.

Sample Space

The list of possible outcomes that could result from the event or experiment. This could be displayed as a list, table, tree diagram or Venn diagram. The notation for sample space as a list is S=\{ ... \}.

A rectangle labelled with an S for sample space, like an empty Venn diagram. The entire rectangle is shaded blue to suggest that everything in the rectangle is in the sample space.

For example, the sample space for tossing a coin is \{ \text{Heads}, \text{Tails}\}.

Event

A set of possible outcomes resulting from an experiment. In general, an event, A, is any subset of a sample space, S.

A Venn diagram where the outer rectangle is labelled S, and the Circle inside the rectangle is labelled A and inside the circle is shaded blue.

For example, if we choose "Heads" for a coin toss then we can say A=\{\text{Heads}\}

Complement of an event

The event that contains all the outcomes in the sample space that are not in the original event.

The complement of event A can be written as A\rq, A^c, \overline{A} or \sim A.

A Venn diagram where the outer rectangle is labelled S, and the Circle inside the rectangle is labelled A. The area inside the rectangle but outside the circle is shaded blue.

If event A \rq contains all the outcomes that are in the sample space S, but not in event A. This means the probability of choosing an outcome from S that is either in A or A\rq is certain or 1. So:

\begin{aligned} P(A) + P(A\rq) &=1 \\ P(A\rq) &=1-P(A) \end{aligned}
\displaystyle P(A\rq) =1-P(A)
\bm{A}
is an event
\bm{A\rq}
is the complement of event A
Intersection

The intersection of events A and B contains all the outcomes that are in both events.

The notation for the intersection of events A and B is A \cap B.

A Venn diagram where the outer rectangle is labelled S, and two intersecting circle inside the rectangle are labelled A and B. The area where the circles overlap is shaded blue.
Union

The union of events A and B contains all the outcomes that are in either or both events.

The notation for the union of events A and B is A \cup B.

A Venn diagram where the outer rectangle is labelled S, and two intersecting circle inside the rectangle are labelled A and B. All of the area inside the two circles is shaded blue.

A subset of an event is a list of outcomes that also exist in the event. If events A and B are subsets of a sample space, S, then A\rq, A \cup B, and A \cap B are also subsets of S.

Worked examples

Example 1

Detemine the sample space for the following experiments:

a

Tossing two coins.

Approach

When flipping a single coin, two outcomes are possible, heads and tails, which can be written as: S=\left\{\text{H}, \text{T}\right\}.

For two coins, there are two heads and two tails, we want every possible combination.

Solution

The sample space can be written as, S = \{ \text{HH}, \text{TT}, \text{HT}, \text{TH} \}

Reflection

We could also display the sample space as a table:

\text{H}\text{T}
\text{H}\text{HH}\text{HT}
\text{T}\text{TH}\text{TT}

Or as a tree diagram:

A tree diagram with two levels. There are two choices at the first level: Heads (H) and Tails (T), each of these branch out to the same two choices: Heads (H) and Tails (T).
b

Selection of a whole number given the interval [2,8].

Approach

The numbers from 2 to 8 are in the closed interval so we should include all the values from 2 to 8.

Solution

Therefore, the sample space for the given interval is:

S=\{2, 3, 4, 5, 6, 7, 8\}

Reflection

If the interval was open, (2,8) we would not include 2 or 8 in the sample space.

c

Drawing an Ace from a standard deck of cards.

Approach

A standard deck has 52 cards with 4 aces, one for each suit: Clubs, Spades, Hearts and Diamonds, as shown below:

A standard deck of 52 cards with 2 black suits, clubs and spades, and 2 red suits, hearts and diamonds. Each suit has 13 cards: Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, and 2.

Solution

The sample space of choosing an Ace from a standard deck of cards is

S=\{\text{Ace of Clubs, Ace of Spades, Ace of Hearts, Ace of Diamonds}\}

Example 2

Consider S which contains the whole numbers from 1 to 20.

A = \left\{3, 6, 9, 12, 15, 18 \right\}\text{ and } B = \left\{1, 2, 3, 4, 5, 6, 7, 8, 9\right\} are events in the sample space S.

a

Find A \cup B.

Approach

The notation A \cup B means the union of A and B. It is the set of elements that are in A or B or both.

Solution

A \cup B = \{ 1,2 ,3 ,4, 5, 6, 7, 8, 9, 12, 15, 18 \}

Reflection

Since this is a set, we do not double count the numbers that are in both events, so we only list 3, 6, 9 once.

b

Find A \cap B.

Approach

The notation A \cap B means the intersection of A and B. It is the set of elements that are in both A and B.

Solution

A \cap B = \{3, 6, 9 \}

c

Find A'.

Approach

The complement of an event are all outcomes that are NOT the event. So we need to include all the elements in S=\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20\}, that are not in A= \left\{3, 6, 9, 12, 15, 18 \right\}.

Solution

A'=\{1,2,4,5,7,8,10,11,13,14, 16,17,19,20\}

d

Find \left(A \cup B\right)'.

Approach

The notation \left(A \cup B\right)' means all outcomes that are not in either A or B but are in S.

From part (a): A \cup B = \{ 1,2 ,3 ,4, 5, 6, 7, 8, 9, 12, 15, 18 \}.

Solution

\left(A \cup B\right)' = 10, 11, 14, 16, 17,19, 20

Reflection

The brackets in the notation (A \cup B)\rq are important, because (A \cup B)\rq \neq A \cup B \rq.

(A \cup B)\rq is the complement of A \cup B, whereas A \cup B\rq is the union of A and B\rq.

Example 3

A card is selected from a standard deck of 52 cards. A is the event of drawing a heart and B is the event of drawing a face card. Write the probability statement and determine the probability of the following events:

a

Selecting a heart and a face card (K, Q, J).

Approach

The probability of drawing a heart AND a face card is the intersection of the probabilities of A and B. This can be written as

P\left(A \cap B\right)

A standard deck has 52 cards with 13 hearts and 12 face cards, as shown below:

A standard deck of 52 cards with 2 black suits, clubs and spades, and 2 red suits, hearts and diamonds. Each suit has 13 cards: Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, and 2.

Solution

First we need to identify the subset A\cap B:A\cap B=\{\text{King of Hearts},\text{Queen of Hearts},\text{Jack of Hearts}\}

Now that we know the total number of outcomes is 3 we can write the solution as the probability \dfrac{3}{52}.

b

Selecting a heart or a face card.

Approach

The probability of drawing a heart OR a face card is the union of the probabilities of A and B. This can be written as P\left(A \cup B\right).

A standard deck has 52 cards with 13 hearts and 12 face cards, as shown below:

A standard deck of 52 cards with 2 black suits, clubs and spades, and 2 red suits, hearts and diamonds. Each suit has 13 cards: Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, and 2.

Solution

First we need to identify the subset A\cup B by listing all cards that are either a Heart, a face card, or both:

A\cup B=\{\text{Ace of Hearts}, \text{King of Hearts}, \text{Queen of Hearts}, \text{Jack of Hearts}, 10 \text{ of Hearts}, 9\text{ of Hearts}, 8\text{ of Hearts}, 7\text{ of Hearts}, 6\text{ of Hearts}, 5\text{ of Hearts}, 4\text{ of Hearts}, 3\text{ of Hearts}, 2\text{ of Hearts}, \text{King of Diamonds}, \text{Queen of Diamonds}, \text{Jack of Diamonds}, \text{King of Spades}, \text{Queen of Spades}, \text{Jack of Spades}, \text{King of Clubs}, \text{Queen of Clubs}, \text{Jack of Clubs}\}

Now that we know the total number of outcomes is 22 we can write the probability as the fraction \dfrac{22}{52}.

Simplified, the probability of drawing a heart or a face card is \dfrac{11}{26}.

Reflection

We do not count any card twice, even if it is an outcome of both events, because it only exists in the deck once.

Outcomes

MA.912.DP.4.1

Describe events as subsets of a sample space using characteristics, or categories, of the outcomes, or as unions, intersections or complements of other events.

MA.912.DP.4.9

Apply the addition and multiplication rules for counting to solve mathematical and real-world problems, including problems involving probability.

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