Problem 53 - Algebra and Trigonometry

Question

What is the probability of rolling “snake eyes” (double ones) three times in a row with a pair of dice?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{46656} \).

Step 1: Understanding the Event

When you roll a pair of dice, "snake eyes" means that both dice show a one. We first need to determine the probability of this event happening in a single roll of two dice.

Step 2: Probability of Rolling Snake Eyes Once

Each die has 6 faces. The probability of rolling a 1 on one die is \( \frac{1}{6} \). Since the two dice rolls are independent, the probability of both dice showing a 1 is \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \).

Step 3: Probability of Repeating the Event Three Times

To find the probability of an event occurring multiple times in succession, we multiply the probability of the event by itself for each occurrence. Thus, the probability of rolling "snake eyes" three times in a row is \( \left(\frac{1}{36}\right)^3 \).

Step 4: Calculating the Final Probability

Compute \( \left(\frac{1}{36}\right)^3 = \frac{1}{36} \times \frac{1}{36} \times \frac{1}{36} = \frac{1}{46656} \).

Step 5: Conclusion

The final probability of rolling "snake eyes" three times in a row with a pair of dice is \( \frac{1}{46656} \).

Key Concepts

Independent Events

When we talk about independent events in probability theory, we mean that the outcome of one event does not affect the outcome of another. For example, with dice probability, the result of one roll doesn’t influence the other.
Each die is rolled separately, and one die showing a particular number doesn't change the likelihood of what appears on the other die. This concept is essential in our original exercise, where we need to find the probability of both dice showing a specific number (1, in this case) simultaneously.

• Each die has 6 faces, and the result on one die is independent of the result on the other die.
• This independence implies that when you roll two dice, the outcomes are not linked in any way.

Understanding independent events allows us to find combined probabilities by multiplying the individual probabilities, as we'll explore in the next sections.

Probability Calculation

Probability is essentially about calculating the chance of a particular outcome occurring, based on the total possible outcomes. To calculate probability, we use the formula: \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). In the context of our dice question, each die has 6 faces, so rolling a particular number, like a '1', is \( \frac{1}{6} \).

When calculating the probability of both dice showing a '1', we assume the events are independent, as discussed. This means we can multiply the probabilities of each die to get the combined probability. So, \( \frac{1}{6} \times \frac{1}{6} = \frac{1}{36} \), for a single occurrence.

• For a repeated independent event, multiply the probability of the single event by itself as many times as needed.
• In our problem, we want triple "snake eyes", so we calculate \( \left(\frac{1}{36}\right)^3 \).

The final calculation becomes \( \frac{1}{36} \times \frac{1}{36} \times \frac{1}{36} = \frac{1}{46656} \), demonstrating the very small likelihood of this precise sequence.

Dice Probability

Dice probability revolves around understanding the likelihood of different outcomes when rolling dice. Standard six-sided dice have faces numbered from 1 to 6, and each face has an equal chance of appearing in any roll.

In our exercise, "snake eyes" refers to both dice showing the number 1. The probability for any specific outcome with one die is \( \frac{1}{6} \). So for two dice simultaneously showing 'snake eyes', the probability is \( \frac{1}{36} \), as each roll is independent.

To grasp dice probability:

• Consider that each die is independent. The combination of results is a simple multiplication of individual event probabilities.
• When determining probability for events like rolling "snake eyes" several times in succession, multiply the single-event probability by itself, considering each success.

Dice games often illustrate probability theory principles since they require understanding both individual outcomes and combined probabilities of multiple rolls. Understanding these basics enriches everything from games to more complex statistical studies.