Problem 10 - Algebra 2

Question

Two standard number cubes are tossed. State whether the events are mutually exclusive. Explain your reasoning. The sum is a prime number; the sum is less than 4

Step-by-Step Solution

Answer

The events 'the sum is a prime number' and 'the sum is less than 4' are not mutually exclusive because there are outcomes that fulfill both conditions simultaneously.

Step 1: Identify Possible Outcomes

First, identify the different possible outcomes when two dice are rolled. There are 36 possible outcomes in total: (1,1), (1,2), (1,3), ... , up to (6,6).

Step 2: Identify Occurrences of Event 1

Identify all situations where the sum is a prime number. A prime number is a number that has only two positive divisors: 1 and itself. The prime numbers less than 12 (the maximum sum of two dice) are 2, 3, 5, 7, and 11. Identify all pairs that sum to these values: {(1,1), (1,2), (2,1), (1,4), (4,1), (2,3), (3,2), (1,6), (6,1), (2,5), (5,2), (3,4), (4,3), (1,5), (5,1), (2,6), (6,2), (3,5), (5,3), (4,4), (1,3), (3,1), (2,4), (4,2), (3,3)}.

Step 3: Identify Occurrences of Event 2

Identify all situations where the sum is less than 4. This includes the pairs {(1,1), (1,2), (2,1), (1,3), (3,1)}.

Step 4: Compare Occurrences

Compare the occurrences of Events 1 and 2. Notice that there are pairs present in both events: (1,1), (1,2), (2,1), (1,3), (3,1).

Step 5: Determine Mutual Exclusivity

Since there are pairs that are included in both events, the two events are not mutually exclusive. For example, when the dice results are (1, 2) or any of the other overlapping pairs, both Event 1 (sum is a prime number) and Event 2 (sum is less than 4) occur at the same time.

Key Concepts

Probability

Probability is the measure of the likelihood that an event will occur. In the context of rolling two dice, understanding probabilities involves looking at the possible outcomes. When you roll a pair of standard six-sided dice, each die can land on a number between 1 and 6. This results in 6 x 6 = 36 possible combinations. To calculate the probability of an event, you divide the number of favorable outcomes by the total number of possible outcomes. For example, to find the probability of rolling a sum of 7, you identify the pair combinations that result in this sum. They are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1) – a total of 6 combinations. Therefore, the probability of rolling a sum of 7 is 6/36, which simplifies to 1/6 or about 16.67%. Probability helps us predict the likelihood of various possible outcomes when dealing with uncertain situations like games involving dice.

Prime Numbers

Prime numbers are fundamental in mathematics. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. This means prime numbers cannot be divided evenly by any other numbers. Examples include 2, 3, 5, 7, and 11. When working with dice, the sum of the numbers on the two dice is sometimes prime. For example, consider the outcomes that result in a prime sum:

• (1, 2) and (2, 1) have a sum of 3
• (1, 4) and (4, 1) have a sum of 5
• (2, 3) and (3, 2) have a sum of 5
• (1, 6) and (6, 1), (2, 5) and (5, 2), and (3, 4) and (4, 3) all have a sum of 7
• (5, 6) and (6, 5) have a sum of 11

To determine if rolling a sum that is prime is possible, one must first identify the numbers that are prime within the potential sums, which in this exercise are numbers less than or equal to 12 – the largest possible sum of two dice.

Sum of Two Dice

Rolling two dice results in a sum that ranges from 2 to 12. Each sum corresponds to one or more dice roll outcomes. Understanding the possible sums is essential in determining probabilities in dice-based games. The sum of two dice can be calculated by adding up the numbers that each die shows. Here are some key points about the sums:

• The smallest sum occurs when both dice show 1, resulting in a sum of 2: (1, 1).
• The largest sum occurs when both dice show 6, resulting in a sum of 12: (6, 6).
• Some sums are achieved with multiple combinations. For instance, a sum of 8 can result from: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2).
• Different sums have different probabilities due to varying combinations that can achieve those sums. For example, a sum of 7 is the most likely because there are six combinations that result in 7.

Understanding these sums allows players to anticipate potential outcomes and decide their strategies in games or solving probability problems.