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Discrete Mathematics: An Active Approach to Mathematical Reasoning

Jennifer Firkins Nordstrom

Section 9.1 Probability

Let \(S\) be a set of possible outcomes, called a sample space. Let \(E\) be a subset of \(S\) with some property. We call \(E\) an event.
The probability of event \(E\text{,}\) \(P(E)\text{,}\) is the number of outcomes in \(E\) divided by the total number of outcomes in \(S\text{.}\)
Let \(N(E)\) be the number of elements in \(E\text{.}\) Similarly, \(N(S)\text{,}\) is the number of outcomes in \(S\text{.}\)
In notation, \(P(E)=\frac{N(E)}{N(S)}\text{.}\)

Example 9.1.1. Finding Probability of an Event.

A standard deck of playing cards has 52 cards. Each card has a suit and a value. The deck has four suits: hearts, diamonds, spades, clubs. Each suit has 13 values: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. The spades and clubs are called “black” cards, the hearts and diamonds are called “red” cards.
Find the probability you randomly choose a club from the deck.
Note, \(E\) is the set of clubs, \(S\) is the set of cards.
Answer 1.
\(P(E)=\frac{13}{52}=\frac{1}{4}\)
Find the probability you randomly choose a red card from the deck.
Note, \(E\) is the set of hearts and diamonds, \(S\) is the set of cards.
Answer 2.
\(P(E)=\frac{26}{52}=\frac{1}{2}\)
Find the probability you randomly choose a 2 from the deck.
Note, \(E\) is the set of 2’s, \(S\) is the set of cards.
Answer 3.
\(P(E)=\frac{4}{52}=\frac{1}{13}\)

Activity 9.1.1.

Suppose you toss a fair coin two times.

(a)

List all the possible outcomes.

(b)

What is the probability that you get exactly 2 heads?

(c)

What is the probability that you get exactly 1 head?
In order to calculate a probability, we need to be able to count equally-likely events. Just because there are, say, 3 outcomes, that doesn’t mean all the outcomes are equally-likely. For example, the Linfied football team could win, lose, or tie a game. Given the team’s winning streak, the three outcomes are not equally-likely. They have a much higher probability of winning than just 1/3.
Thus, the focus for the rest of the chapter will be on counting events. Once we can count outcomes, it is straight-forward to find the probability.
Before moving on, though, we will state a few useful facts about probabilities.
  • \(P(E)\) is always a number between 0 and 1. The probability can be 0.5. It cannot be 50%
  • \(P(E)=0\) means the event is not possible.
  • \(P(E)=1\) means the event always happens.

Reading Questions Check Your Understanding

1.

In a standard deck of cards, find the probability of randomly drawing an Ace. Give your answer to at least 3 decimal places.
The probability is .

2.

In a standard deck of cards, find the probability of randomly drawing a face card (Jack, Queen, King). Give your answer to at least 3 decimal places.
The probability is .

3.

In a standard deck of cards, find the probability of randomly drawing an even numbered card. Give your answer to at least 3 decimal places.
The probability is .

4.

An urn contains 3 red balls, 2 green balls, 4 multicolored balls. Find the probability of drawing a green ball. Give your answer to at least 3 decimal places.
The probability is .

5.

An urn contains 3 red balls, 2 green balls, 4 multicolored balls. Find the probability of drawing a solid colored ball. Give your answer to at least 3 decimal places.
The probability is .

Exercises Exercises

1.

Assume the sample space is a standard deck of 52 cards. Suppose you choose a random card from the deck.
  1. List all the possible outcomes of the card being red and not a face card.
  2. Calculate the probability of the event in (a).

2.

Assume the sample space is a standard deck of 52 cards. Suppose you choose a random card from the deck.
  1. List all the possible outcomes of the card being black and having an even number on it.
  2. Calculate the probability of the event in (a).

3.

Assume the sample space is the possible rolls of a pair of dice, one blue die and one red die.
  1. List all the possible outcomes where the sum of the numbers showing face up is 8.
  2. Calculate the probability of the event in (a).

4.

Assume the sample space is the possible rolls of a pair of dice, one blue die and one red die.
  1. List all the possible outcomes where the sum of the numbers showing face up is at most 6.
  2. Calculate the probability of the event in (a).

5.

Suppose a coin is tossed four times and the side showing face up on each toss is noted. Suppose also that heads and tails are equally-likely for the coin.
  1. List the 16 elements in the sample space as sequences of heads and tails.
  2. List the set of outcomes and the probability for the event of exactly one toss resulting in a head.
  3. List the set of outcomes and the probability for the event that at least two tosses result in a head.
  4. List the set of outcomes and the probability for the event that no toss results in a head.

6.

Suppose that on a true/false exam you have no idea about the answers to three questions. You choose your answers randomly and therefore have a 50-50 chance of being correct on any one question. For example, let \(CCW\) indicate that you were correct on the first two questions but wrong on the third.
  1. List all the elements of the sample space as the possible sequences of \(C\) and \(W\) for the three questions.
  2. List the set of outcomes and the probability for the event that exactly one answer is correct.
  3. List the set of outcomes and the probability for the event that at least two answers are correct.
  4. List the set of outcomes and the probability for the event that no answers are correct.
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