Problem 22 - A First Course in Probability

Question

A red die, a blue die, and a yellow die (all six sided) are rolled. We are interested in the probability that the number appearing on the blue die is less than that appearing on the yellow die, which is less than that appearing on the red die. That is,with \(B, Y,\) and \(R\) denoting, respectively, the number appearing on the blue, yellow, and red die, we are interested in \(P(B

Step-by-Step Solution

Answer

The overall probability of \(P(B

Step 1: a) Probability that no two of the dice land on the same number

First, we need to determine the total number of possible outcomes when rolling three six-sided dice. This can be calculated as 6 (number of sides) * 6 * 6 = 216. Next, we will calculate the number of possible outcomes when no two dice have the same number. We have 6 choices for the value of the first die, then 5 choices for the second die (since it cannot have the same value as the first), and 4 choices for the third die (since it must be different from the first two). This gives us a total of 6 * 5 * 4 = 120 different outcomes. Now to find the probability of this event, we will divide the number of outcomes of interest (120) by the total number of possible outcomes (216): \(P(\text{no two dice land on the same number}) =\frac{120}{216} = \frac{20}{36} = \frac{5}{9}\)

Step 2: b) Conditional probability that B < Y < R

Since we know that no two dice have the same number, there are 3! = 6 ways in which they can be ordered. Out of these 6 possible orderings, only 1 follows the pattern B < Y < R. So the conditional probability that B < Y < R given that no two dice land on the same number is: \(P(B

Step 3: c) Overall probability P(B < Y < R)

To find the overall probability of P(B < Y < R), we can multiply the probability of each event happening: \(P(B

Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already happened. This is a fundamental concept in probability theory and helps us understand relationships between different random events.
For example, in this exercise, we're interested in finding the probability that the number rolled on the blue die is less than the number on the yellow die, which in turn is less than the number on the red die. However, this comes with a condition: no two dice should land on the same number.
To compute this conditional probability, we use the formula:

• Determine the probability of the condition event (no two dice having the same number), which is \( \frac{5}{9}\). This was calculated by dividing the desirable outcomes by the total possible outcomes when rolling three dice.
• Next, find the number of permutations that satisfy the constraint (B < Y < R). Since each of the three dice shows a different number and only one permutation – (B, Y, R) – satisfies the condition among 3! = 6 permutations, we have \( \frac{1}{6}\) for this probability.

Thus, by multiplying these two probabilities, we obtain the conditional probability: \( P(B < Y < R) = \frac{5}{54}\).

Combinatorics

Combinatorics is the branch of mathematics that studies finite or countable discrete structures. In simpler terms, it deals with the counting, arrangement, and combination of objects.

In our case, combinatorics allows us to systematically organize our approach to solving probability problems with dice. By rolling three different colored dice, we're interested in the number of ways they can be ordered from smallest to largest without repeating any number.

• The total permutations for three different numbers is calculated using factorials (3! = 6). This captures all the orders in which these numbers can appear on the dice.
• We'll use combinatorics to help break the problem down into more manageable parts, which is essential for calculating probability, especially with conditions.

Understanding and applying the basics of combinatorics is crucial for tackling more complex probability problems efficiently and accurately.

Dice Probabilities

Dice probabilities are a classic example when studying probability theory. With dice, each face has an equal chance of showing, assuming the dice are fair.

When calculating the probability of events with dice, consider:

• Total outcomes: Each die has 6 faces, so rolling one die results in 6 outcomes. For three dice, you calculate total outcomes as 6 * 6 * 6 = 216.
• Desired outcomes for specific probability calculations are determined based on the problem requirements. In our exercise, we're looking for unique outcomes where no two dice show the same number, which was calculated as 120 outcomes.

To find the probability of specific events involving dice, it's essential to clearly define the conditions and the number of favorable outcomes. Dice problems often introduce randomness that can be tackled by logical breakdown and methodical calculations of possibilities and outcomes.

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Problem 21

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Problem 23

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Key Concepts

Conditional Probability

Conditional probability is the likelihood of an event occurring given that another event has already happened. This is a fundamental concept in probability theory and helps us understand relationships between different random events.
For example, in this exercise, we're interested in finding the probability that the number rolled on the blue die is less than the number on the yellow die, which in turn is less than the number on the red die. However, this comes with a condition: no two dice should land on the same number.
To compute this conditional probability, we use the formula:

• Determine the probability of the condition event (no two dice having the same number), which is \( \frac{5}{9}\). This was calculated by dividing the desirable outcomes by the total possible outcomes when rolling three dice.
• Next, find the number of permutations that satisfy the constraint (B < Y < R). Since each of the three dice shows a different number and only one permutation – (B, Y, R) – satisfies the condition among 3! = 6 permutations, we have \( \frac{1}{6}\) for this probability.

Thus, by multiplying these two probabilities, we obtain the conditional probability: \( P(B < Y < R) = \frac{5}{54}\).

Combinatorics

Combinatorics is the branch of mathematics that studies finite or countable discrete structures. In simpler terms, it deals with the counting, arrangement, and combination of objects.

In our case, combinatorics allows us to systematically organize our approach to solving probability problems with dice. By rolling three different colored dice, we're interested in the number of ways they can be ordered from smallest to largest without repeating any number.

• The total permutations for three different numbers is calculated using factorials (3! = 6). This captures all the orders in which these numbers can appear on the dice.
• We'll use combinatorics to help break the problem down into more manageable parts, which is essential for calculating probability, especially with conditions.

Understanding and applying the basics of combinatorics is crucial for tackling more complex probability problems efficiently and accurately.

Dice Probabilities

Dice probabilities are a classic example when studying probability theory. With dice, each face has an equal chance of showing, assuming the dice are fair.

When calculating the probability of events with dice, consider:

• Total outcomes: Each die has 6 faces, so rolling one die results in 6 outcomes. For three dice, you calculate total outcomes as 6 * 6 * 6 = 216.
• Desired outcomes for specific probability calculations are determined based on the problem requirements. In our exercise, we're looking for unique outcomes where no two dice show the same number, which was calculated as 120 outcomes.

To find the probability of specific events involving dice, it's essential to clearly define the conditions and the number of favorable outcomes. Dice problems often introduce randomness that can be tackled by logical breakdown and methodical calculations of possibilities and outcomes.