Problem 19 - A First Course in Probability

Question

Two symmetric dice have both had two of their sides painted red, two painted black, one painted yellow, and the other painted white. When this pair of dice is rolled, what is the probability that both dice land with the same color face up?

Step-by-Step Solution

Answer

The probability that both dice land with the same color face up is \(\frac{5}{18}\).

Step 1: Find the total number of outcomes

When a die has 6 sides, there are 6 possible outcomes when it is rolled. So when two dice are rolled together, the total number of outcomes will be 6 * 6, which is 36 possible outcomes.

Step 2: Find the total number of favorable outcomes

Let's find out all the options in which both dice show the same color. There are 4 colors: Red, Black, Yellow and White. Each color has a different number of faces on each die: - Red: 2 faces - Black: 2 faces - Yellow: 1 face - White: 1 face Now let's examine each color and find the number of favorable outcomes for each color pair: 1) Both dice have Red face up: There are 2 Red faces on each die, so there are \(2 * 2 = 4\) outcomes. 2) Both dice have Black face up: There are 2 Black faces on each die, so there are \(2 * 2 = 4\) outcomes. 3) Both dice have Yellow face up: There is 1 Yellow face on each die, so there is \(1 * 1 = 1\) outcome. 4) Both dice have White face up: There is 1 White face on each die, so there is \(1 * 1 = 1\) outcome. We add all these favorable outcomes to get the total number of favorable outcomes: \(4 + 4 + 1 + 1 = 10\).

Step 3: Calculate the probability

Now that we have the total number of favorable outcomes (10) and the total number of outcomes (36), we can find the probability by dividing the favorable outcomes by the total outcomes. Probability of both dice showing the same color face up = \(\frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{10}{36}\) To further simplify the fraction, we can divide both the numerator and the denominator by their greatest common divisor, which is 2: \(= \frac{10}{36} ÷ \frac{2}{2} = \frac{5}{18}\) So the probability that both dice land with the same color face up is \(\frac{5}{18}\).

Key Concepts

Dice Probability

When you roll a die, you're engaging with one of the oldest concepts of probability — Dice Probability. Each die has six faces, hence, when you roll one die, there are 6 possible outcomes. Things become more intriguing when rolling two dice. The number of possible outcomes in this scenario multiplies, as each die's face acts independently of the other.

• For a single die, there are 6 outcomes (each face).
• For two dice, the outcomes are calculated by multiplying the number of outcomes for one die by the number of both dice: 6 * 6 = 36 outcomes.

This foundation of calculating outcomes is crucial not just for dice, but many probability problems. Here, we apply this understanding to find not just the possible, but favorable outcomes where both dice show the same color face up.

Combinatorics

Combinatorics is a branch of mathematics dealing with counting, both as a means and as an end in obtaining results, and deciding when to count. In our exercise, combinatorics helps us understand how to count favorable outcomes — outcomes where both dice show the same color. For each color on the dice, we analyze how many faces correspond to the same color: - Red: 2 faces - Black: 2 faces - Yellow: 1 face - White: 1 face Detailed counting for each color using combinatorics gives: - **Both dice Red:** 2 Red faces on each die lead to 2 * 2 = 4 outcomes. - **Both dice Black:** Similarly, 2 * 2 = 4 outcomes. - **Both dice Yellow:** 1 face on each means 1 * 1 = 1 outcome. - **Both dice White:** Again, 1 * 1 = 1 outcome. Summing these counts gives us 10 favorable outcomes from the distinct color pairs, essential for calculating probabilities.

Outcomes in Probability

Understanding outcomes in probability is central to solving problems like having two dice showing the same color. Outcomes in probability refer to the possible results that can occur from a probabilistic event.In this context:

• Total number of outcomes: When rolling two dice, there are 36 total possible outcomes (6 from each die).
• Favorable outcomes: As calculated with combinatorics, there are 10 outcomes where both dice show the same color.

The probability is then the ratio of favorable outcomes to the total outcomes:\[\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{10}{36}\]After simplifying, this becomes: \[\frac{5}{18}\]This fraction represents the chance that both dice land with the same color face up, and it highlights how effective counting and simplification can reveal these probabilities.