Problem 1 - Calculus for Biology and Medicine

Question

Toss a fair coin twice. Let \(X\) be the random variable that counts the number of tails in each outcome. Find the probability mass function describing the distribution of \(X\).

Step-by-Step Solution

Answer

The PMF is \( P(X=x) = \{\frac{1}{4}, \frac{1}{2}, \frac{1}{4}\} \) for \( x=0, 1, 2 \).

Step 1: List All Possible Outcomes

A fair coin has two possible outcomes when tossed: heads (H) or tails (T). For two tosses, list all possible combinations: HH, HT, TH, TT. These represent the sample space S of this probability experiment.

Step 2: Define the Random Variable X

Let the random variable \(X\) represent the number of tails in the outcome of the two coin tosses. Assign values to \(X\) based on the outcomes: X = 0 (HH), X = 1 (HT, TH), X = 2 (TT).

Step 3: Calculate Probabilities for X = 0

Count the outcomes that result in 0 tails, which is the outcome HH. There is 1 outcome, so the probability \( P(X=0) = \frac{1}{4} \).

Step 4: Calculate Probabilities for X = 1

Count the outcomes that result in 1 tail, which are HT and TH. There are 2 such outcomes, so the probability \( P(X=1) = \frac{2}{4} = \frac{1}{2} \).

Step 5: Calculate Probabilities for X = 2

Count the outcomes that result in 2 tails, which is the outcome TT. There is 1 outcome, so the probability \( P(X=2) = \frac{1}{4} \).

Step 6: Write the Probability Mass Function (PMF)

The probability mass function of \( X \) is given by: \[ P(X=x) = \begin{cases} \frac{1}{4}, & \text{if } x = 0, \ \frac{1}{2}, & \text{if } x = 1, \ \frac{1}{4}, & \text{if } x = 2. \end{cases} \] This function describes the distribution of the number of tails obtained.

Key Concepts

Understanding Random Variables

In probability and statistics, a random variable is a function that assigns a numerical value to each outcome in a sample space of a random experiment. Think of it as a way to "count" or "measure" certain outcomes in a mathematical way.

Here are the key aspects of random variables:

• They represent real-world events numerically, providing a bridge between random experiments and numerical analysis.
• Random variables can be discrete (having distinct values) or continuous (having a range of values).
• In our exercise, the random variable \(X\) is discrete and counts the number of tails in the two coin tosses.

By defining \(X\), we can analyze the probability of different outcomes based on our coin toss scenario. This is a central concept for exploring probability distributions.

Understanding Coin Toss

A coin toss is one of the simplest examples of a random experiment. It's used frequently in probability theory because it provides a clear, binary outcome: heads (H) or tails (T). This characteristic makes it ideal for learning basic probability concepts.

Let's break down the coin toss experiment:

• A single coin toss has two possible outcomes, making it a binomial experiment.
• When tossing a fair coin, each outcome (H or T) has an equal probability of \(\frac{1}{2}\).
• By considering two tosses, we expand the experimentation, forming pairs like HH, HT, TH, TT.

With multiple tosses, as seen in our exercise, you can see how probabilities combine to affect outcomes, building a more complex sample space.

Exploring Probability Distribution

Probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. Specifically, the probability mass function (PMF) gives the probability that a discrete random variable is exactly equal to some value.

Key points about probability distributions in this context:

• For discrete random variables like \(X\), the PMF summarizes the likelihood of different outcomes within the sample space.
• In our coin toss experiment, the PMF details probabilities for \(X = 0\), \(X = 1\), and \(X = 2\), based on observed outcomes.
• The PMF shows that the probabilities add up to 1, which validates the total coverage of all possible events.

By understanding probability distributions, you can better analyze and predict the results of random experiments, like how likely certain numbers of tails appear in two coin tosses.