Lesson 1 Probability and Counting

Motivating Example

In 1620, the Grand Duke of Tuscany asked the astronomer and mathematician Galileo to determine whether the numbers on three fair dice are more likely to add up to 9 or 10.

Theory

Definition 1.1 (Definition of Probability) If all possible outcomes are equally likely, then the probability of an event \(A\) is \[ P(A) = \frac{\text{number of outcomes where $A$ occurs}}{\text{number of possible outcomes}}. \]

For example, suppose we want to know the probability of getting an even number when we roll a fair die. There are \(6\) equally likely possible outcomes,

⚀ ⚁ ⚂ ⚃ ⚄ ⚅,

of which 3 are even. Therefore, the probability of rolling an even number is \[ P(\text{even number}) = \frac{\text{number of outcomes that are even}}{\text{number of possible outcomes}} = \frac{3}{6}. \]

Next, suppose we want to know the probability of rolling a 7 when we roll two fair dice. There are 36 equally likely outcomes,

⚀⚀	⚁⚀	⚂⚀	⚃⚀	⚄⚀	⚅⚀
⚀⚁	⚁⚁	⚂⚁	⚃⚁	⚄⚁	⚅⚁
⚀⚂	⚁⚂	⚂⚂	⚃⚂	⚄⚂	⚅⚂
⚀⚃	⚁⚃	⚂⚃	⚃⚃	⚄⚃	⚅⚃
⚀⚄	⚁⚄	⚂⚄	⚃⚄	⚄⚄	⚅⚄
⚀⚅	⚁⚅	⚂⚅	⚃⚅	⚄⚅	⚅⚅

of which 6 outcomes add up to 7. (Check this for yourself!) Therefore, the probability of rolling a 7 is \[ P(\text{sum is 7}) = \frac{\text{number of outcomes where the sum is 7}}{\text{number of possible outcomes}} = \frac{6}{36}. \]

Galileo needed to consider the possible outcomes when three dice are rolled. At this point, listing all of the possible outcomes is impractical. We need to find a way to count the number of outcomes without listing all of them.

Here is a way to see that there are 36 ways to roll two dice, without listing them out. There are 6 possible outcomes for the first die, and each of these 6 outcomes can be paired with 6 possible outcomes for the second die. Therefore, there must be \(6 \cdot 6 = 36\) ways to roll two dice. The following theorem generalizes this principle.

Theorem 1.1 (Multiplication Principle of Counting) If a task can be performed in \(n_1\) ways, and for each of these ways, a second task can be performed in \(n_2\) ways, then the two tasks can be performed together in a total of \(n_1 \cdot n_2\) ways.

Proof. List the outcomes in a table, with \(n_1\) columns and \(n_2\) rows (like we did above for the possible outcomes of two dice, where \(n_1 = n_2 = 6\)). Each column corresponds to one of the \(n_1\) possible outcomes of the first task, each row to one of the \(n_2\) possible outcomes of the second. There are \(n_1 \cdot n_2\) cells in the table.

Using the multiplication principle, we can count the total number of ways of rolling three dice. We already know that there are 36 ways that the first two dice can come out. Each of these 36 ways can be matched with each of the 6 ways that the third dice can come out. Therefore, by Theorem @ref{thm:multiplication-principle}, there are \[ 36 \cdot 6 = 216 \] ways the three dice could come out.

Using the multiplication principle, we can calculate the probability that no sixes are rolled among the three dice. In order for there to be no sixes, each of the three dice must have shown one of the other 5 numbers. The number of ways for this to happen is \[ 5 \cdot 5 \cdot 5 = 125, \] so the probability is \[ P(\text{no sixes}) = \frac{125}{216} \approx 57.8\%. \]

Now, Galileo was interested in the event that the numbers add up to 9, which is much trickier to count. We will come back to his question in a future lesson, once we have more counting tools under our belt.

Examples

1. In the casino game craps, two dice are rolled by a player, called the “shooter”. The first roll is called the “come-out roll”. The shooter wins on the come-out roll if they roll a 7 or a 11. On the other hand, they lose if the come-out roll is a 2, 3, or 12. Otherwise, the game continues.
   1. What is the probability that the shooter wins on the come-out roll?
   2. What is the probability that the shooter loses on the come-out roll?
   3. What is the probability that the game continues?
   4. If you add up the three probabilities you just calculated, what do you get? Why does this make sense?
2. Many examples in this book will be based on a standard deck of 52 playing cards, consisting of 13 cards of each suit. Familiarize yourself with the terminology. Suppose you draw one card from a shuffled deck of cards.
   1. What is the probability that the card is a heart?
   2. What is the probability that the card is an ace?
   3. What is the probability that the card is a face card (also known as a “court card”)?
3. Many examples in this book will be based on the casino game roulette. Familiarize yourself with the roulette wheel. (Click the wheel to spin it.) You win if the ball lands in a pocket that matches your bet.
   1. What is the probability that you win if you bet on red?
   2. What is the probability that you win if you bet on the number 10?
   3. What is the probability that you win if you bet on odd (i.e., that the number is odd)?
4. In standard poker, each player is dealt 5 cards off the top of a (shuffled) deck of cards. The hand is called a “flush” if all 5 cards are the same suit. What is the probability that you get a flush of hearts (i.e., all 5 cards are hearts)?

Additional Exercises