Question

A dice rolled twice, what is the probability that the two dice show a different number? (1) \(2 / 3\) (2) \(1 / 6\) (3) \(5 / 6\) (4) \(1 / 2\)

Short answer

Answer: The probability that the two dice show different numbers is \(\frac{5}{6}\).

Step-by-step solution

Determine the number of favorable outcomes

To count the favorable outcomes, we need to find the number of combinations where the two dice show different numbers. There are 6 possible outcomes for the first dice roll. Since the second dice roll must be different from the first, it will have 5 options because one number is already chosen by the first dice. Therefore, there are 6 x 5 = 30 favorable outcomes where the two dice show different numbers.

Calculate the total number of possible outcomes

Aside from the 30 favorable outcomes, we identify that we have a total of 36 possible outcomes when rolling the dice twice as discussed previously (6 possible outcomes for the first dice roll and 6 possible outcomes for the second dice roll, which results in 6 x 6 = 36 total outcomes).

Calculate the probability

To find the probability of the two dice showing different numbers, we divide the number of favorable outcomes by the total number of possible outcomes: Probability = (Number of favorable outcomes) / (Total number of possible outcomes) Plug in the numbers we found in Steps 1 and 2: Probability = 30 / 36 Simplify the fraction: Probability = \(\frac{5}{6}\) The probability that the two dice show a different number is 5/6. The correct answer is (3) \(5 / 6\).

Key concepts

Favorable Outcomes

When we talk about "favorable outcomes" in probability, we mean the specific outcomes that satisfy the conditions of a given problem. In this scenario, we need the outcomes where two dice, rolled one after the other, display different numbers. Here is how you can think about it:

1. Roll the first die. There are 6 possible numbers it can land on, as each face has a different number from 1 to 6.
2. Now, for the second die, we want it to show a number different from the first. This leaves us with only 5 possible numbers to choose from, as one number is already used by the first die.
3. Therefore, the number of favorable outcomes is the result of multiplying the outcomes of the first die with possible outcomes for the second die: 6 (for the first) × 5 (for the second).

Hence, the total favorable outcomes are 30 combinations. This describes every set where two dice show distinct numbers.

Total Possible Outcomes

"Total possible outcomes" refers to all potential ways the dice can land, regardless of the specific conditions of interest. When rolling two dice consecutively, each die has 6 faces, so the calculations are fairly straightforward.

1. Roll the first die - it has 6 possible outcomes (1 through 6).
2. Roll the second die - it also has 6 different outcomes, totally independent of the first.

This means the total possible outcomes from rolling two dice is equal to 6 (for the first die) multiplied by 6 (for the second die), giving us a total of 36 combinations.

It provides a broad view of all the arrangements possible when two dice are rolled, forming the basis against which we compare favorable outcomes to determine probability.

Dice Combinations

Understanding "dice combinations" is crucial because it allows us to visualize and compute probabilities. When a single die is rolled, there are straightforward possibilities: 1 through 6. However, when two dice are involved, the combinations grow quickly.

Let's break it down with an example:

• Rolling a 1 on the first die pairs with 6 possibilities on the second die: (1,1), (1,2), (1,3), and so on up till (1,6).
• Next is rolling a 2 on the first die, which can also pair with 6 variations on the second die: (2,1) through (2,6).
• This pattern repeats for each result from 1 to 6 on the first die.

While this is an explanation for all possible outcomes, each unique pair constitutes one of the 36 total combinations aside from our specific interest in favorable outcomes, where the numbers differ.

This understanding helps in visualizing how dice behave in terms of outcomes and forms the base for calculating probabilities using combinations.