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 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 (B < Y < R) is \(5/54\).

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

There are a total of 6*6*6 = 216 possible outcomes when rolling three dice. To find the number of outcomes where no two dice land on the same number, let's consider all the possibilities with each die rolled: - The blue die can land on any number from 1 to 6 (6 choices) - The yellow die cannot land on the same number as the blue die, so it has 5 choices (6 - 1 = 5). - The red die cannot land on the same number as the blue or yellow dice, so it has 4 choices (6 - 2 = 4). Multiplying the choices together, we find that there are 6 * 5 * 4 = 120 outcomes where no two dice land on the same number. Thus, the probability of this event is 120/216, which simplifies to 20/36 or 5/9.

Step 2: (b) Conditional probability that B < Y < R given no two dice land on the same number

Now that we know the probability of no two dice landing on the same number, we can consider the conditional probability of B < Y < R. Since the numbers on each die are distinct and no two are the same, the three dice can be arranged in 3! (3 factorial) ways, which is 6 different orders. Out of these 6 possibilities, only one of them (B < Y < R) meets our condition. Thus, the conditional probability of B < Y < R given that no two dice land on the same number is 1/6.

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

To calculate the overall probability that B < Y < R, we need to multiply the probability of no two dice landing on the same number (5/9) by the conditional probability of B < Y < R given that no two dice land on the same number (1/6). This gives us: \(P(B

Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory, reflecting the likelihood of an event occurring given that another event has already occurred. In our dice example, we first calculated the probability of each die rolling a distinct number without considering the order. Once we established that distinct outcomes are achieved, we then asked for the probability of the blue die rolling a number less than the yellow die, which in turn is less than the number of the red die. This is a classic example of conditional probability because we're looking at a subset of the original possibilities—in this case, the event where the dice roll distinct numbers—and asking for the chances of a particular order occurring within that subset. The crucial part is understanding that the first condition of distinct numbers must be met before we can consider the sequential order of B < Y < R.

Factorial

The factorial function, denoted by an exclamation point (!), is integral to calculating probabilities dealing with distinct arrangements or permutations. It's the product of all positive integers up to a given number. For instance, 3! (read as 'three factorial') is equal to 3 x 2 x 1, which is 6. This comes into play in part (b) of our exercise because once we've established that each die shows a distinct number, there are 3! ways to arrange these three numbers. Since there's only one arrangement out of these that follows the pattern of B < Y < R (blue less than yellow less than red), the factorial helps us swiftly calculate the conditional probability by showcasing the different possible arrangements we can make out of a set number of distinct items.

Distinct Outcomes

In probability, distinct outcomes are the result of an event where each outcome is unique and does not repeat. When rolling three six-sided dice, for example, we might refer to 'distinct outcomes' as those rolls where no two dice land on the same number. The computation for part (a) of the problem revolves around calculating the number of ways we can get distinct outcomes from the three dice. We systematically reduce the number of possibilities for each subsequent die, ensuring no repetition occurs. This concept is critical because the probability of successive events often relies on their outcomes being distinct, and it forms the foundation of more complex probability scenarios.

Permutations

Permutations relate to the number of ways in which objects can be arranged when order matters. Each unique arrangement is called a permutation. When we look at dice rolls where the numbers are distinct, there are several permutations for how those numbers can come up. For 3 dice with distinct outcomes, like in our dice example, the permutations are found using factorial notation—3!. This represents all the ways we can arrange three distinct items, which is particularly handy when calculating probabilities for sequences or orders of events. By understanding permutations, we can tackle more sophisticated probability challenges where the order or arrangement of outcomes is a crucial element of the scenario.

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Problem 23

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Key Concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory, reflecting the likelihood of an event occurring given that another event has already occurred. In our dice example, we first calculated the probability of each die rolling a distinct number without considering the order. Once we established that distinct outcomes are achieved, we then asked for the probability of the blue die rolling a number less than the yellow die, which in turn is less than the number of the red die. This is a classic example of conditional probability because we're looking at a subset of the original possibilities—in this case, the event where the dice roll distinct numbers—and asking for the chances of a particular order occurring within that subset. The crucial part is understanding that the first condition of distinct numbers must be met before we can consider the sequential order of B < Y < R.

Factorial

The factorial function, denoted by an exclamation point (!), is integral to calculating probabilities dealing with distinct arrangements or permutations. It's the product of all positive integers up to a given number. For instance, 3! (read as 'three factorial') is equal to 3 x 2 x 1, which is 6. This comes into play in part (b) of our exercise because once we've established that each die shows a distinct number, there are 3! ways to arrange these three numbers. Since there's only one arrangement out of these that follows the pattern of B < Y < R (blue less than yellow less than red), the factorial helps us swiftly calculate the conditional probability by showcasing the different possible arrangements we can make out of a set number of distinct items.

Distinct Outcomes

In probability, distinct outcomes are the result of an event where each outcome is unique and does not repeat. When rolling three six-sided dice, for example, we might refer to 'distinct outcomes' as those rolls where no two dice land on the same number. The computation for part (a) of the problem revolves around calculating the number of ways we can get distinct outcomes from the three dice. We systematically reduce the number of possibilities for each subsequent die, ensuring no repetition occurs. This concept is critical because the probability of successive events often relies on their outcomes being distinct, and it forms the foundation of more complex probability scenarios.

Permutations

Permutations relate to the number of ways in which objects can be arranged when order matters. Each unique arrangement is called a permutation. When we look at dice rolls where the numbers are distinct, there are several permutations for how those numbers can come up. For 3 dice with distinct outcomes, like in our dice example, the permutations are found using factorial notation—3!. This represents all the ways we can arrange three distinct items, which is particularly handy when calculating probabilities for sequences or orders of events. By understanding permutations, we can tackle more sophisticated probability challenges where the order or arrangement of outcomes is a crucial element of the scenario.