Problem 3 - Calculus for Biology and Medicine

Question

Roll a fair die twice. Let \(X\) be the random variable that gives the absolute value of the differences between the two numbers. Find the probability mass function describing the distribution of \(X\).

Step-by-Step Solution

Answer

The probability mass function is: \(P(X=0) = \frac{1}{6}\), \(P(X=1) = \frac{5}{18}\), \(P(X=2) = \frac{2}{9}\), \(P(X=3) = \frac{2}{9}\), \(P(X=4) = \frac{1}{6}\), \(P(X=5) = \frac{1}{18}\).

Step 1: List all possible outcomes

When a fair die is rolled twice, there are 36 possible outcomes. Each outcome can be represented as an ordered pair \((a, b)\), where \(a\) and \(b\) range from 1 to 6.

Step 2: Calculate the random variable

Define the random variable \(X\) as the absolute difference between the two numbers: \(X = |a - b|\). This will take values from 0 to 5 depending on \(a\) and \(b\).

Step 3: Count outcomes for each value of X

Calculate the number of outcomes for each possible value of \(X\):- \(X = 0\): Occurs when \(a = b\). There are 6 such outcomes (1,1), (2,2), ..., (6,6).- \(X = 1\): Occurs when \(|a-b| = 1\). Count pairs like (1,2), (2,1), etc. There are 10 such outcomes.- \(X = 2\): Similarly, count (1,3), (3,1), etc. There are 8 outcomes.- \(X = 3\): Pairs like (1,4), (4,1), etc. also result in 8 outcomes.- \(X = 4\): Similar counting gives us 6 outcomes.- \(X = 5\): Only pairs (1,6) and (6,1) result in 2 outcomes.

Step 4: Determine probabilities

To find the probability mass function, divide the number of outcomes for each \(X\) by the total number of outcomes (36):- \(P(X=0) = \frac{6}{36} = \frac{1}{6}\)- \(P(X=1) = \frac{10}{36} = \frac{5}{18}\)- \(P(X=2) = \frac{8}{36} = \frac{2}{9}\)- \(P(X=3) = \frac{8}{36} = \frac{2}{9}\)- \(P(X=4) = \frac{6}{36} = \frac{1}{6}\)- \(P(X=5) = \frac{2}{36} = \frac{1}{18}\)

Key Concepts

Understanding the Random Variable

In probability and statistics, a random variable is a fundamental concept. It represents a numerical outcome derived from a stochastic process, like rolling a die. Here, the random variable is typically denoted by a capital letter, such as \(X\). In the exercise mentioned, \(X\) stands for the absolute difference between the numbers rolled on two dice.A key aspect of random variables:

• They can take on various values, each associated with a probability.
• They summarize the results of a random event numerically.

By understanding the random variable in this context, we can express the process of rolling two dice mathematically. If you roll a fair die twice, the random variable \(X\) takes on the value \(|a-b|\), where \(a\) and \(b\) are the numbers shown on the first and second die, respectively. Knowing the possible ranges of \(X\) is essential to mapping its distribution, setting the stage for understanding the probability mass function.

The Role of Absolute Difference

Absolute difference is a measure that gives a non-negative value, representing the magnitude of difference between two numbers, without considering which is larger.When calculating the absolute difference, it's simply the value of the subtraction between two numbers, without the negative sign. In this die-rolling exercise:

• For any numbers rolled, \(a\) and \(b\), the absolute difference is \(|a-b|\).
• This approach ensures that we only get possible values from 0 to 5, which encompasses all differences possible when rolling two six-sided dice.

Understanding absolute difference in the context of random variables is vital. It dictates the potential results of \(X\). Specifically, the smallest possible absolute difference is 0 (occurring when \(a = b\)), while the largest absolute difference is 5 (for pairs like (1,6) or (6,1)). This clear range of outcomes forms the base for calculating the probability distribution of \(X\).

Probability and Die Rolling

Rolling dice is a classic example of a random experiment, which serves as a great illustration of probability concepts.When you roll a fair six-sided die, each side has an equal chance of landing face up, with a probability of \(\frac{1}{6}\). When two dice are rolled:

• There are 36 different ordered outcomes, encompassing all pairs \((a, b)\) where \(a\) and \(b\) each range from 1 to 6.
• The outcomes of \(X\), as calculated by the absolute difference, range from 0 to 5.

The probability mass function (pmf) represents the likelihood of each potential value of \(X\). By dividing the number of outcomes for each \(X = |a-b|\) by the total number of outcomes (36), you can find the probability for each difference value.Here's a quick recap of these probabilities:

• \(P(X=0) = \frac{1}{6}\)
• \(P(X=1) = \frac{5}{18}\)
• \(P(X=2) = \frac{2}{9}\)
• \(P(X=3) = \frac{2}{9}\)
• \(P(X=4) = \frac{1}{6}\)
• \(P(X=5) = \frac{1}{18}\)

This distribution shows how likely it is to have each particular absolute difference when rolling a pair of dice, demonstrating how probability theory elegantly captures real-world randomness.