Question

You roll a fair die three times. What is the probability that a) you roll all 6 's? b) you roll all odd numbers? c) none of your rolls gets a number divisible by 3 ? d) you roll at least one 5 ? e) the numbers you roll are not all \(5^{\prime}\) s?

Short answer

a) \(\frac{1}{216}\), b) \(\frac{1}{8}\), c) \(\frac{8}{27}\), d) \(\frac{91}{216}\), e) \(\frac{215}{216}\).

Step-by-step solution

Determine the Sample Space

A fair die has 6 outcomes: 1, 2, 3, 4, 5, or 6. When rolling a die three times independently, the total number of possible outcomes is given by: \[ 6 \times 6 \times 6 = 6^3 = 216 \]. Thus, the sample space consists of 216 possible outcomes.

Calculate Probability of Rolling All 6's

To roll all 6's, each of the three rolls must result in a 6. There is only 1 favorable outcome: (6, 6, 6). Therefore, the probability is:\[ P(\text{all 6's}) = \frac{1}{216} \].

Calculate Probability of Rolling All Odd Numbers

The odd numbers on a die are 1, 3, and 5. Thus, there are 3 favorable outcomes for each roll. Therefore, the total number of favorable sequences for three rolls is: \[ 3 \times 3 \times 3 = 27 \]. The probability is given by:\[ P(\text{all odd}) = \frac{27}{216} = \frac{1}{8} \].

Calculate Probability of No Numbers Divisible by 3

Numbers divisible by 3 on a die include 3 and 6. Numbers not divisible by 3 are 1, 2, 4, and 5. Thus, there are 4 favorable outcomes for each roll. The total number of favorable sequences for three rolls is:\[ 4 \times 4 \times 4 = 64 \]. The probability is:\[ P(\text{no numbers divisible by 3}) = \frac{64}{216} = \frac{8}{27} \].

Calculate Probability of Rolling at Least One 5

Instead, calculate the probability of not rolling any 5's and subtract from 1. The probability of not rolling a 5 for one roll is \(\frac{5}{6}\). Thus, for three rolls, that probability is:\[ \left(\frac{5}{6}\right)^3 = \frac{125}{216} \]. Therefore, the probability of rolling at least one 5 is:\[ P(\text{at least one 5}) = 1 - \frac{125}{216} = \frac{91}{216} \].

Calculate Probability of Not All 5's

If we roll all 5's, we have 1 favorable outcome, (5, 5, 5). Thus, the probability of rolling all 5's is:\[ \frac{1}{216} \]. Therefore, the probability of not rolling all 5's is:\[ P(\text{not all 5's}) = 1 - \frac{1}{216} = \frac{215}{216} \].

Key concepts

Sample Space

The concept of **sample space** is quite fundamental in probability theory. It refers to the set of all possible outcomes in a probability experiment. For instance, when you roll a fair six-sided die, each face represents a possible outcome - namely 1 through 6. Furthermore, when you perform the experiment of rolling this die three times, the sample space becomes more extensive.

In this scenario, each roll is independent, meaning any number from 1 to 6 can appear on every single roll without affecting the others.

• This independence means we have six options for the first roll, six for the second, and six for the third.
• Thus, the size of the sample space for three independent die rolls is calculated as follows: \[ 6 imes 6 imes 6 = 216 \]Thus, we have 216 different possible outcomes.

Die Roll Outcomes

Understanding the **die roll outcomes** is crucial for calculating probabilities. Each time you roll a die, you can get one of six results: 1, 2, 3, 4, 5, or 6. When considering multiple rolls:

• Repeated rolls can result in repeated numbers. For instance, (2, 4, 4) is a possible outcome among others.
• The significance of each outcome becomes evident when considering specific questions. For example, rolling all 6's and the sequence (6, 6, 6) is just one specific outcome within the sample space.

Knowing the exact outcomes allows you to isolate and categorize them as favorable or not, depending on what probability scenario you are investigating.

Favorable Outcomes

When calculating probabilities, recognizing **favorable outcomes** is your first step towards solving the problem. A favorable outcome is a specific result or set of results that align with the event in question.

• For example, in the scenario where we are interested in rolling all odd numbers, the favorable outcomes are sequences like (1, 3, 5), (1, 1, 1), etc., that only consist of odd numbers - precisely 27 such combinations exist.
• Calculating favorable outcomes requires understanding what constitutes a success in a given experiment and then counting the number of occurrences in which this success appears.

Favorable outcomes form the basis for the probability calculation itself, found by \[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} \] Understanding this concept allows us to gauge how likely a particular event is to occur within the confines of the sample space.

Independent Events

In probability, **independent events** are ones where the outcome of one event does not influence the outcome of another. When rolling a die, each roll is independent of any others.

• This means rolling a 1, the first time, does not affect the probability of rolling a 1, a 2, or any other numbers for subsequent rolls.
• Mathematically, two events A and B are independent if and only if: \[ P(A \cap B) = P(A) \times P(B) \]Here, \( \cap \) denotes the intersection, or the probability of both events occurring.

In our context of rolling a die three times, each result remains unaffected by the previous roll, maintaining a consistent probability distribution throughout the experiment. Such independence simplifies many calculations, allowing for easier combination of probabilities across several events.