Problem 5 - Discrete Mathematics with Applications

Question

Two dice are rolled. Find the probability of obtaining: Two fives.

Step-by-Step Solution

Answer

The probability of obtaining two fives when rolling two dice is \(\frac{1}{36}\).

Step 1: Find the total possible outcomes

When rolling two dice, there are 6 sides on each die, which gives us 6 possible outcomes for each die. Since we have two dice, the total possible outcomes will be the 6 outcomes of the first die multiplied by the 6 outcomes of the second die. This can be represented as: \(6 \times 6 = 36\)

Step 2: Find the favorable outcomes

The favorable outcome in this case is rolling two fives. There is only one way to achieve this, which is when both dice have a five as their outcome. So, the number of favorable outcomes is 1.

Step 3: Calculate the probability

In order to find the probability of the favorable outcome, we need to divide the number of favorable outcomes by the total possible outcomes. This can be represented by: \(\frac{Number\,of\,favorable\,outcomes}{Total\,possible\,outcomes}\) In this case, we have: \(\frac{1}{36}\) So, the probability of obtaining two fives when rolling two dice is \(\frac{1}{36}\).

Key Concepts

Understanding the Probability of Two Dice

In probability theory, when we talk about rolling two dice, we are examining the different ways two dice can land. Each of these dice has 6 sides, and each side represents one outcome. When two dice are rolled, each die acts independently of the other, meaning the result of one does not affect the result of the other.
This independence allows us to calculate the total possible outcomes by multiplying the number of outcomes for the first die by the number of outcomes for the second die.

• For a single die, there are 6 possible outcomes: 1, 2, 3, 4, 5, and 6.
• With two dice, the calculation is: \( 6 \times 6 = 36 \)

This gives us 36 unique combinations of numbers when two dice are rolled. Grasping this basic principle is crucial for solving any probability problem involving multiple dice.

Favorable and Total Outcomes

To compute the probability of an event, we need to distinguish between favorable outcomes and total outcomes. Total outcomes are all the possible combinations that can occur when two dice are thrown. In our previous calculation, we found there were 36 total outcomes.
Favorable outcomes, on the other hand, are specific outcomes that match the event we are interested in—rolling two fives in this case.
Since each die must show a five, only one specific combination of outcomes is favorable:

• Die 1 and Die 2 both show a 5, which is \((5,5)\).

Therefore, the number of favorable outcomes is 1. By understanding the difference between these, we can move onto probability calculations.

Probability Calculation Steps

Calculating probability follows a straightforward process using the formula: \[\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\]To find the probability of rolling two fives, substitute the values we've determined:

• The number of favorable outcomes is 1 (just the \((5,5)\) outcome).
• The total number of outcomes is 36.

This gives the probability as: \[\frac{1}{36}\]This outcome represents a very rare event since only 1 of the 36 possible combinations results in double fives. By following this method, you can calculate probabilities for any event related to rolling two dice.