Rolling the Dice: Finding the Probability of At Most One Six

Suppose three fair dice are rolled. What is the probability at most one six appears?

Probability is a fundamental concept in mathematics, and it plays an important role in many fields such as statistics, physics, and finance. When we roll a die, the outcome is random, and the probability of getting a specific number is 1/6. But what happens when we roll multiple dice? In this blog post, we will explore the probability of at most one six appearing when rolling three fair dice.

To find the probability of at most one six appearing, we need to consider all the possible outcomes and count the ones that meet our condition. Let's start by finding the total number of outcomes when rolling three dice. Since each die has six possible outcomes, the total number of outcomes for three dice is 6 x 6 x 6 = 216.

Next, we need to count the number of outcomes where at most one six appears. There are four cases to consider:

No sixes: In this case, we can choose any number from 1 to 5 for each of the three dice. There are 5 x 5 x 5 = 125 ways to do this.

One six: There are three ways to choose which die will show a six, and five ways to choose the value of each of the other two dice (1, 2, 3, 4, or 5). Therefore, there are 3 x 5 x 5 = 75 ways to get exactly one six.

Two sixes: There are three ways to choose which dice will show a six, and one way to choose the value of the other die. Therefore, there are 3 x 1 x 5 = 15 ways to get exactly two sixes.

Three sixes: There is only one way to get three sixes.

Adding up the number of outcomes for each case, we get 125 + 75 + 15 + 1 = 216 outcomes that meet our condition.

Therefore, the probability of at most one six appearing when rolling three fair dice is 216/216 = 1, or 100%.

We calculated the probability of at most one six appearing when rolling three fair dice. We found that the probability is 100%, which means that it is guaranteed that at most one six will appear. This result may seem surprising, but it is a consequence of the fact that we are rolling three dice, and the probability of rolling three sixes is very low. The concept of probability is a powerful tool that allows us to make predictions and decisions based on the likelihood of different outcomes.