Problem 27 - Calculus for Biology and Medicine

Question

Roll two fair dice, one after the other, and find the probability that the first number is larger than the second number.

Step-by-Step Solution

Answer

The probability is \( \frac{5}{12} \).

Step 1: Determine All Possible Outcomes

When rolling two dice, each die has 6 faces. Therefore, the total number of possible outcomes when rolling two dice is obtained by multiplying the number of outcomes for each die: \( 6 \times 6 = 36 \).

Step 2: List Favorable Outcomes

We need the outcomes where the first number is larger than the second number. List all the combinations for each face of the first die: if the first die shows 2, possible outcomes are (2,1). For 3, outcomes are (3,1), (3,2), and so on up to when the first die shows 6 for which outcomes are (6,1), (6,2), (6,3), (6,4), and (6,5).

Step 3: Count Favorable Outcomes

The number of favorable outcomes for each value of the first die is as follows: 1 possibility when the first die is 2, 2 possibilities when it's 3, 3 when it's 4, 4 when it's 5, and 5 when it's 6. Summing gives \( 1 + 2 + 3 + 4 + 5 = 15 \) favorable outcomes.

Step 4: Calculate Probability

The probability that the first number is larger than the second number is the number of favorable outcomes divided by the total number of possible outcomes: \( \frac{15}{36} \). Simplifying gives \( \frac{5}{12} \).

Key Concepts

Favorable Outcomes

When dealing with probability, the term "favorable outcomes" is crucial. It refers to the specific results we are interested in when conducting an experiment or a random event. In the context of rolling two dice, the favorable outcomes are the scenarios where the first number rolled is greater than the second number.
Take, for example, when the first die shows a 2. The only favorable outcome is (2, 1). As you progress to a 3 on the first die, the favorable outcomes increase to (3, 1) and (3, 2). This continues up to when the first die shows a 6, where possible outcomes range from (6, 1) to (6, 5).
This systematic approach to identifying favorable outcomes helps in visualizing how often our desired event occurs and forms the basis for calculating probabilities.

Total Possible Outcomes

Understanding the total possible outcomes is one of the fundamental steps in calculating probability. When rolling two dice, each die has 6 faces, leading to a combination of outcomes.

To determine the total outcomes, multiply the number of outcomes for the first die by those of the second die. Hence, it would be:

• First die: 6 possible outcomes (1 through 6)
• Second die: 6 possible outcomes (1 through 6)

The total possible outcomes thus become: \[ 6 \times 6 = 36 \]Each outcome corresponds to one of the possible pair wise results of rolling these two dice. Represents every potential permutation of outcomes, allowing us to assess the probability of any set of favorable outcomes.

Dice Rolls

Rolling dice is a classic experiment in probability theory, often used to demonstrate the principles of random events and calculations of probability. Each die has identical faces, numbered from one to six, creating simple, yet numerous combinations when two dice are involved.
Here are a few helpful things to remember about dice rolls:

• Each die rolls independently of the other. The outcome of one does not affect the other.
• Each number on a die is equally likely to occur due to the fairness of the die.

When you roll two dice one after the other, the permutations of results are elaborate yet methodical, as seen with the exercise. This makes it easier to systematically write out and count the outcomes to find probabilities. This methodical approach to dice rolls not only aids in understanding favorable and total possible outcomes, it also serves as an ideal introduction to more complex probability problems.