Problem 35 - Calculus for Biology and Medicine

Question

Roll a fair die six times. Let \(X\) be the number of times you roll a 6 . Find the probability mass function.

Step-by-Step Solution

Answer

The probability mass function is \( P(X = k) = \binom{6}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\right)^{6-k} \) for \( k = 0, 1, 2, 3, 4, 5, 6 \).

Step 1: Define the Random Variable

The random variable \( X \) represents the number of times a 6 is rolled when a fair six-sided die is rolled six times. So \( X \) can take values from 0 to 6.

Step 2: Recognize the Distribution

Since we are counting the number of successes (rolling a 6) in a fixed number of trials (6 rolls), this scenario follows a binomial distribution.

Step 3: Determine Parameters of the Binomial Distribution

For a binomial distribution, we have \( n = 6 \) (number of trials) and \( p = \frac{1}{6} \) (probability of rolling a 6 on a single roll).

Step 4: Write the Probability Mass Function (PMF) Formula

The PMF of a binomial distribution is given by: \[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]where \( \binom{n}{k} \) is the binomial coefficient.

Step 5: Substitute Known Values into PMF

Substitute \( n = 6 \) and \( p = \frac{1}{6} \) into the PMF formula:\[P(X = k) = \binom{6}{k} \left(\frac{1}{6}\right)^k \left(\frac{5}{6}\right)^{6-k}\]This formula gives the probability for each possible value of \( k \), from 0 to 6.

Key Concepts

Random Variable

In probability and statistics, a random variable represents a numerical outcome of a random phenomenon. For our exercise, the random variable, denoted as \( X \), counts the number of times a 6 is rolled when a fair six-sided die is rolled six times.

Because we are seeking the number of occurrences of a specific outcome (rolling a 6), \( X \) is a discrete random variable, meaning it can only take on a limited set of values. In this case, \( X \) can be any whole number from 0 (if no 6s were rolled) to 6 (if every roll resulted in a 6). By defining a random variable in this context, we are able to mathematically describe possible outcomes of our experiment.

Understanding random variables is a foundational step in calculating probabilities, as it helps us systematically assess and compute likelihoods based on observed or expected results.

Probability Mass Function

A probability mass function (PMF) provides the probabilities of a discrete random variable. It tells us how likely each individual outcome is for a random event. For a binomially distributed random variable, like our \( X \), the PMF can be expressed using a specific formula.

The PMF for our situation is:

• \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Here, \( \binom{n}{k} \) is a binomial coefficient, \( p \) represents the probability of success on a single trial, and \( n \) is the number of trials.

In this die-rolling exercise, the PMF formula allows us to calculate the probability of rolling exactly \( k \) sixes out of six trials:

• "Success" is defined as rolling a 6, so \( p = \frac{1}{6} \).
• There are \( n = 6 \) rolls in total.
• \( k \) varies from 0 to 6, representing the possible number of sixes.

By substituting these values into the PMF, we can determine the probabilities for each potential outcome.

Binomial Coefficient

The binomial coefficient is a key component in calculating binomial probabilities. It determines how many ways we can choose \( k \) successes from \( n \) independent trials and is written as \( \binom{n}{k} \).

The binomial coefficient is calculated using the formula:

• \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

In this formula, \( n! \) (read as "n factorial") means multiplying all positive integers up to \( n \), and the same applies for \( k! \) and \((n-k)!\). This counts the different possible arrangements of \( k \) specific outcomes within \( n \) trials, given that order does not matter.

For example, if you want to find \( \binom{6}{2} \), you calculate:

• \[ \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15 \]

This means there are 15 combinations of getting exactly 2 sixes when the die is rolled six times. The binomial coefficient is a cornerstone of understanding how combinations work in probability theory, particularly in contexts like the binomial distribution.