Problem 11 - Algebra for College Students

Question

If a pair of dice is tossed, then what is the probability of getting a) a pair of 2’s? b) at least one 2? c) a sum of 7? d) a sum greater than 1? e) a sum less than 2?

Step-by-Step Solution

Answer

a) \(\frac{1}{36}\), b) \(\frac{11}{36}\), c) \(\frac{1}{6}\), d) 1, e) 0.

Step 1: Total number of outcomes

Each die has 6 faces. When rolling two dice, the total number of possible outcomes is \(6 \times 6 = 36\).

Step 2: Probability of getting a pair of 2’s

There is only one outcome where both dice show a 2, which is (2,2). Thus, the probability is \(\frac{1}{36}\).

Step 3: Probability of getting at least one 2

First, calculate the probability of not getting a 2 on either die. Each die has 5 faces that are not 2. Therefore, the probability of not getting a 2 on one die is \(\frac{5}{6}\). The probability of not getting a 2 on either die is \(\left(\frac{5}{6}\right)^2 = \frac{25}{36}\). The probability of getting at least one 2 is the complement: \(1 - \frac{25}{36} = \frac{11}{36}\).

Step 4: Probability of getting a sum of 7

The possible outcomes that sum to 7 are (1,6), (2,5), (3,4), (4,3), (5,2), (6,1). There are 6 such outcomes. Thus, the probability is \(\frac{6}{36} = \frac{1}{6}\).

Step 5: Probability of getting a sum greater than 1

The sum of the numbers must be at least 2. Since all the sums are possible (from 2 to 12) in a fair roll, the probability is \(\frac{36}{36} = 1\).

Step 6: Probability of getting a sum less than 2

The smallest possible sum when rolling two dice is 2 (1+1). There are no possible outcomes for a sum less than 2, so the probability is \(\frac{0}{36} = 0\).

Key Concepts

Probability Calculation

Let's dive into how we calculate the probabilities using the example of rolling two dice. Each die has 6 faces, so when rolling two dice, we have a total of 36 possible outcomes. Now, if we want to find the probability of a specific event happening, we simply divide the number of favorable outcomes by the total number of possible outcomes. For example, if we want to find the probability of rolling a pair of 2’s, there’s only one favorable outcome: (2,2). Thus, the probability is \( \frac{1}{36} \). This straightforward division approach forms the basis of probability calculation.

Outcomes in Probability

In probability, understanding outcomes is crucial. An outcome is a possible result of a probabilistic event. For two dice, there are 36 possible outcomes because each die has 6 faces. These outcomes can be listed as pairs: (1,1), (1,2), ..., (6,6). Let’s see this in action with examples from the provided exercise:
- For a sum of 7, the favorable outcomes are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Therefore, there are 6 outcomes that add up to 7, and the probability is \( \frac{6}{36} = \frac{1}{6} \).
- If we want a sum greater than 1, any outcome is possible because the smallest sum with two dice is 2. Hence, the probability is \( \frac{36}{36} = 1 \) or certainty.
- For a sum less than 2, no possible outcomes exist since the minimum sum is 2 (1 + 1). So, the probability is \( \frac{0}{36} = 0 \).

Complement Rule in Probability

The complement rule in probability helps us find the likelihood of an event not happening. The probability of the complement of an event is given by 1 minus the probability of the event. Let's apply this to our exercise:
- To find the probability of rolling at least one 2 on either die, we first calculate the probability of rolling no 2s. Each die has a \( \frac{5}{6} \) chance of not showing a 2. So, for both dice, the probability is \( \left( \frac{5}{6} \right)^2 = \frac{25}{36} \).
- Using the complement rule, the probability of getting at least one 2 is \( 1 - \frac{25}{36} = \frac{11}{36} \). This rule simplifies calculations, especially when dealing with probabilities of 'at least' scenarios.