Question

Predicting compositions of independent events. Suppose you roll a fair six- sided die three times. (a) What is the probability of getting a 5 twice from all three rolls of the dice? (b) What is the probability of getting a total of at least two 5 's from all three rolls of the die?

Short answer

(a) \( \frac{5}{72} \); (b) \( \frac{2}{27} \)

Step-by-step solution

- Determine the probability of rolling a 5

Since a fair six-sided die has six faces, the probability of rolling a 5 on any single roll is \( \frac{1}{6} \).

- Determine the probability of not rolling a 5

The probability of not rolling a 5 on any single roll is \( 1 - \frac{1}{6} = \frac{5}{6} \).

- Use the Binomial Probability Formula for part (a)

The problem involves a binomial distribution where the number of trials (n) is 3, the number of successes (k) is 2, and the probability of success (p) is \( \frac{1}{6} \). The binomial probability formula is \[ P(X=k) = \binom{n}{k} p^k (1-p)^{n-k} \]. Substituting the values: \[ P(X=2) = \binom{3}{2} \left( \frac{1}{6} \right)^2 \left( \frac{5}{6} \right)^{1} = 3 \cdot \frac{1}{36} \cdot \frac{5}{6} = \frac{15}{216} = \frac{5}{72} \].

- Calculate the probability for part (b)

For part (b), calculate the probability for exactly 2 fives plus the probability for exactly 3 fives. We already have the probability for exactly 2 fives (\( \frac{5}{72} \)). For exactly 3 fives: \[ P(X=3) = \binom{3}{3} \left( \frac{1}{6} \right)^3 \left( \frac{5}{6} \right)^{0} = 1 \cdot \frac{1}{216} \cdot 1 = \frac{1}{216} \]. Adding both probabilities: \[ P(X\geq2) = P(X=2) + P(X=3) = \frac{5}{72} + \frac{1}{216} = \frac{15}{216} + \frac{1}{216} = \frac{16}{216} = \frac{2}{27} \].

Key concepts

Probability Theory

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes. When rolling a fair six-sided die, each face has an equal chance of landing up. This makes calculating probabilities straightforward. For instance, the likelihood of rolling a 5 is \(\frac{1}{6}\). This probability remains constant with every individual roll because each roll is an independent event.
To further understand, imagine rolling the die multiple times. Even though past rolls do not affect future rolls, probability can still help us determine the likelihood of specific combinations occurring across a series of trials. The foundational idea is that the probabilities of all possible outcomes add up to 1.
Probability theory is used in various fields, like games of chance, physics, and even in everyday decision-making processes.

Binomial Distribution

A binomial distribution is crucial for understanding probabilities of outcomes across multiple trials. It deals with scenarios where there are a fixed number of trials, each trial has only two possible outcomes (success or failure), and the probability of success remains the same for each trial.
In our example, rolling a six-sided die three times forms a classic binomial distribution case. We're interested in the probability of getting exactly two 5s in three rolls. Here's where the binomial probability formula comes in handy:
\[ P(X=k) = \binom{n}{k} \, p^k \, (1-p)^{n-k} \]
In this formula, \(n\) is the number of trials, \(k\) is the number of successes, and \(p\) is the probability of success on each trial. Using our example:
\[ P(X=2) = \binom{3}{2} \, \, \left( \frac{1}{6} \right)^2 \, \left( \frac{5}{6} \right)^1 = \frac{5}{72} \]
This result signifies a roughly 6.9% chance of rolling exactly two 5s in three rolls.
Understanding binomial distribution helps in various statistical analyses, from quality control tests in manufacturing to determining success rates in medical treatments.

Statistical Thermodynamics

Statistical thermodynamics is an area bridging statistical mechanics and classical thermodynamics. It explains macroscopic properties of systems based on the statistical behaviors of their microscopic constituents and relies on probabilities and distributions.
The concepts of statistical thermodynamics can sometimes be illustrated through simpler examples like rolling dice. While the direct application of our exercise to thermodynamics might seem a stretch, the underlying mathematics of predicting outcomes and distributions are entirely analogous.
In statistical thermodynamics, we often deal with large numbers of particles and configurations. The probabilities of particles' positions and energies are assessed similarly to how we handle dice roll outcomes. This can offer insights into how energy distributes in a system, how temperatures affect reaction rates, and more.
Though our example of die rolls is simple, understanding probability theory and binomial distribution forms the foundation for more complex analyses, which are essential in fields like thermodynamics, quantum mechanics, and beyond.