Understanding the Probability of Rolling a Number Divisible by 3 on a Die

Probability is a fundamental concept in mathematics, and one of the simplest scenarios to understand involves rolling a die. When a standard six-sided die is thrown, what is the probability of getting a number divisible by 3?

Simple Probability with a Single Die

A standard six-sided die has the numbers 1 through 6. The numbers that are divisible by 3 in this range are 3 and 6. The probability of rolling a number that is divisible by 3 can be calculated as follows:

Counting Favorable Outcomes and Total Outcomes

Favorable Outcomes: There are 2 favorable outcomes: 3 and 6.

Total Outcomes: There are 6 possible outcomes when rolling a die (1, 2, 3, 4, 5, 6).

The probability P is calculated as:

P Number of favorable outcomes / Total number of outcomes 2 / 6 1 / 3

Thus, the probability of getting a number divisible by 3 when throwing a single die is 1/3.

Probability with Two Dice

When rolling two six-sided dice, there are 36 possible outcomes (6 x 6 36). To find the probability of rolling a number that is evenly divisible by 3, we need to count how many of these outcomes result in a number divisible by 3.

The numbers divisible by 3 in the range of 2 to 12 are 3, 6, 9, and 12. We can list the combinations that result in these numbers:

Combinations of Numbers Divisible by 3

3: (1, 2) (2, 1) 6: (1, 5) (2, 4) (3, 3) (4, 2) (5, 1) 9: (3, 6) (4, 5) (5, 4) (6, 3) 12: (6, 6)

Adding these up, there are a total of 12 outcomes that result in a number divisible by 3. Therefore, the probability is:

12 / 36 1 / 3 0.3333

Probability with Multiple Dice

For the sake of completeness, let’s consider the scenario of rolling a pair of six-sided dice and the combinations that result in numbers divisible by 3:

List of Outcomes Divisible by 3

3: (1, 2), (2, 1) 6: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1) 9: (3, 6), (4, 5), (5, 4), (6, 3) 12: (6, 6)

Here, we can see that:

Two 3's Five 6's Four 9's One 12

Adding these up, there are 12 numbers divisible by 3. Thus, the probability is 12 / 36 1 / 3.

General Formula and Probability Notation

Let's define the sample space S as the set of all possible outcomes, and let A be the event that a number is divisible by 3. Here, S {1, 2, 3, 4, 5, 6}, and n(S) 6. The event A {3, 6} gives n(A) 2.

The probability P(A) is given by:

P(A) n(A) / n(S) 2 / 6 1 / 3

This general formula can be applied to any number of dice and any range of numbers, provided they follow the same principles of probability.