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Chapter 13: Problem 39

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 's?

Short Answer

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

Step by step solution

01

Understanding the Experiment

The experiment involves rolling a fair six-sided die three times. The die has sides numbered from 1 to 6. For each roll, the outcome is independent of the previous rolls.
02

Calculating Probability for All 6's (a)

The probability of rolling a 6 on a fair die in a single roll is \( \frac{1}{6} \). For three independent rolls, the probability of rolling 6's each time is: \[ \left( \frac{1}{6} \right)^3 = \frac{1}{216} \]
03

Calculating Probability for All Odd Numbers (b)

The odd numbers on a die are 1, 3, and 5. The probability of rolling an odd number is \( \frac{3}{6} = \frac{1}{2} \). For three independent rolls, the probability of rolling odd numbers each time is: \[ \left( \frac{1}{2} \right)^3 = \frac{1}{8} \]
04

Calculating Probability for No Number Divisible by 3 (c)

Numbers divisible by 3 are 3 and 6. Thus, the numbers not divisible by 3 are 1, 2, 4, and 5. The probability of getting a number not divisible by 3 is \( \frac{4}{6} = \frac{2}{3} \). For three rolls: \[ \left( \frac{2}{3} \right)^3 = \frac{8}{27} \]
05

Calculating Probability for At Least One 5 (d)

First, calculate the probability of not rolling a 5 in a single roll: \( \frac{5}{6} \). The probability of not rolling a 5 in all three rolls is: \( \left( \frac{5}{6} \right)^3 = \frac{125}{216} \). The probability of at least one 5 is: \[ 1 - \frac{125}{216} = \frac{91}{216} \]
06

Calculating Probability for Not All 5's (e)

The probability of rolling all 5's is \( \left( \frac{1}{6} \right)^3 = \frac{1}{216} \). Then, the probability of not rolling all 5's is: \[ 1 - \frac{1}{216} = \frac{215}{216} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Independent Events
In probability, independent events are those whose outcomes do not affect each other. When rolling a die multiple times, each roll is independent because the outcome of one roll does not influence the next. This means:
  • The probability of getting a certain number on one roll is unaffected by what numbers were rolled before.
  • When calculating probabilities for independent events, you multiply the probabilities of individual events.
For example, rolling a six three times involves independent events where the probability of rolling a six on each roll is \(\frac{1}{6}\). Therefore, you multiply this probability for each roll to find the total probability.
Probability Calculation
Probability is the measure of how likely an event is to occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. For a fair six-sided die:
  • The probability of rolling any specific number, such as a six, is \(\frac{1}{6}\) because there is one favorable outcome and six possible outcomes.
  • To find the probability of rolling a number like an odd number, add up all favorable outcomes and divide by the total. Odd numbers on a die are 1, 3, and 5, so the probability is \(\frac{3}{6} = \frac{1}{2}\).
  • If you want the probability of not rolling a number divisible by 3, first count outcomes not divisible by 3 (1, 2, 4, 5) and use \(\frac{4}{6} = \frac{2}{3}\).
When faced with multiple events, these probabilities are often multiplied due to the independence of each event.
Dice Probability
Dice probability explores the chances of outcomes when rolling a die.
  • Each face of a six-sided die has an equal chance of landing face up, which is \(\frac{1}{6}\).
  • When rolling multiple times, each roll is independent, hence you can multiply the probabilities of individual outcomes.
  • For sequences like rolling odd numbers three times, calculate \( \left( \frac{1}{2} \right)^3 \), representing the chance on each of the three rolls.
  • Complementary events like "at least one five" involve calculating the probability of the opposite happening (no fives) and subtracting from one. \(1 - \left( \frac{5}{6} \right)^3\) meters the odds of rolling at least one five.
These concepts help in tackling more complex questions concerning dice probability scenarios.
Combinatorics
Combinatorics deals with counting possibilities and configurations, vital for solving probability problems. In dice rolling:
  • Each roll offers six possibilities, meaning three rolls provide \(6 \times 6 \times 6 = 216\) total combinations.
  • To determine how certain combinations, like all of one number, can occur, reduce possibilities. For example, all fives can occur in only one way (5, 5, 5), leading to \(\frac{1}{216}\).
  • When needing combinations where at least one roll matches a condition, use complementary counting. Calculate the opposite (none match) and subtract from total combinations.
Understanding these principles helps in both simple and complex probability calculations by structuring and recognizing all possible outcomes efficiently.

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