Problem 16 - Algebra and Trigonometry

Question

A pair of dice is rolled, and the numbers showing are observed. (a) List the sample space of this experiment. (b) Find the probability of getting a sum of 7. (c) Find the probability of getting a sum of 9. (d) Find the probability that the two dice show doubles (the same number). (e) Find the probability that the two dice show different numbers. (f) Find the probability of getting a sum of 9 or higher.

Step-by-Step Solution

Answer

(a) 36 outcomes. (b) Probability is \(\frac{1}{6}\). (c) \(\frac{1}{9}\). (d) \(\frac{1}{6}\). (e) \(\frac{5}{6}\). (f) \(\frac{5}{18}\).

Step 1: Understand the Experiment

When rolling a pair of dice, each die has 6 faces with numbers from 1 to 6. Each die is rolled once, and the outcome is a combination of the top faces of the two dice.

Step 2: List the Sample Space

The sample space includes all possible outcomes when two dice are rolled. Each die has 6 faces, resulting in a total of 36 combinations. The sample space is: \[(1,1), (1,2), ..., (1,6), (2,1), ..., (6,6)\].

Step 3: Calculate Probability of a Sum of 7

First, find pairs that add up to 7: \((1,6), (2,5), (3,4), (4,3), (5,2), (6,1)\). There are 6 such pairs. Probability = \(\frac{6}{36} = \frac{1}{6}\).

Step 4: Calculate Probability of a Sum of 9

Identify the pairs that sum to 9: \((3,6), (4,5), (5,4), (6,3)\). There are 4 such pairs. Probability = \(\frac{4}{36} = \frac{1}{9}\).

Step 5: Calculate Probability of Doubles

Find pairs where both dice show the same number: \((1,1), (2,2), (3,3), (4,4), (5,5), (6,6)\). There are 6 such pairs. Probability = \(\frac{6}{36} = \frac{1}{6}\).

Step 6: Calculate Probability of Different Numbers

Subtract the probability of same numbers from 1. Probability of same numbers is \(\frac{1}{6}\), so different numbers probability = \(1 - \frac{1}{6} = \frac{5}{6}\).

Step 7: Calculate Probability of Sum of 9 or Higher

Find pairs with sums 9, 10, 11, and 12: \((3,6), (4,5), (5,4), (6,3), (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)\). Total pairs = 10. Probability = \(\frac{10}{36} = \frac{5}{18}\).

Key Concepts

Sample Space

When you roll a pair of dice, you observe the numbers that appear on the top faces of each die. In probability theory, the set of all possible outcomes of this experiment is known as the sample space. Since each die has 6 faces with numbers ranging from 1 to 6, and you roll two dice, the outcomes include every combination of these numbers. To express the sample space of rolling two dice, we list all numerically distinct pairs:

• (1,1), (1,2), (1,3), ..., (1,6)
• (2,1), (2,2), (2,3), ..., (2,6)
• ...
• (6,1), (6,2), (6,3), ..., (6,6)

Altogether, there are 36 possible outcomes, as there are 6 outcomes for the first die and each of these can occur with any of the 6 outcomes of the second die.
This results in a total sample space containing 36 elements.

Sum of Dice

The sum of dice refers to the total of the numbers appearing on the upper faces of two rolled dice. Understanding the sum of dice is crucial for determining specific probabilities in games of chance that involve dice.
For instance, to find out how likely it is to roll a sum of 7, identify all possible pairs that add up to 7:

• (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

There are 6 such combinations, meaning a 1-in-6 chance, or a probability of \( \frac{1}{6} \).
Similarly, the pairs that add to a sum of 9 are:

• (3,6), (4,5), (5,4), (6,3)

This gives a probability of \( \frac{4}{36} = \frac{1}{9} \).
Understanding these sums directly influences strategic decisions in games involving dice.

Rolling Dice

Rolling dice is a classic random experiment in probability theory. Each roll of a die is independent of others, and each number has an equal chance of appearing on the top face. When rolling two dice simultaneously, each roll of an individual die still maintains this independence.
However, combining them results in more complex outcome consideration since it produces ordered pairs representing each possible outcome.
The exercise asked about:

• Doubles, which occur when both dice show the same number: (1,1), (2,2), ..., (6,6). There are 6 double outcomes, leading to a probability of \( \frac{6}{36} = \frac{1}{6} \).
• Different numbers, calculated as the complement of rolling doubles, resulting in a probability of \( \frac{5}{6} \).

Whether in games or puzzles, knowing the rules that dictate the results of dice rolls guides probability calculations.

Probability Calculations

Probability calculations for dice involve determining the likelihood of various events. Each outcome in the sample space is equally probable; therefore, the probability of any particular event is calculated using the ratio of favorable outcomes to the total number of possible outcomes.
When we look for the probability of achieving a sum greater than or equal to 9, we need to count all combinations that satisfy this condition:

• (3,6), (4,5), (5,4), (6,3) for the sum of 9
• (4,6), (5,5), (6,4) for sum of 10
• (5,6), (6,5) for sum of 11
• (6,6) for sum of 12

That results in a total of 10 outcomes, so the probability is \( \frac{10}{36} = \frac{5}{18} \).
Understanding probability theory helps predict these results and provides insight into the likelihood of outcomes in many real-world scenarios and games.