Problem 2 - Calculus for Biology and Medicine

Question

Toss a fair coin four times. Let \(X\) be the random variable that counts the number of heads. Find the probability mass function describing the distribution of \(X\).

Step-by-Step Solution

Answer

The PMF for the number of heads in 4 tosses is: P(X=0)=0.0625, P(X=1)=0.25, P(X=2)=0.375, P(X=3)=0.25, P(X=4)=0.0625.

Step 1: Understand the Random Variable

We have a fair coin and we are tossing it four times. The random variable \(X\) represents the number of heads that appear in these four tosses. Therefore, \(X\) can take values from \(0\) to \(4\), inclusive.

Step 2: Find Possible Outcomes

Each coin toss has two possible outcomes: heads or tails. Since we toss the coin four times, the total number of outcomes is \(2^4 = 16\).

Step 3: Use Binomial Probability Formula

The situation described can be modeled by a binomial distribution where \(n = 4\) (the number of trials), \(p = 0.5\) (the probability of getting heads in one toss), and \(X\) is the number of successful trials (heads). The probability mass function for a binomial distribution is given by: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] where \(k = 0, 1, 2, 3, 4\).

Step 4: Calculate Probabilities for Each Value of X

Calculate \(P(X=k)\) for each value of \(k\) using the binomial formula: - \(P(X=0) = \binom{4}{0} (0.5)^0 (0.5)^4 = 1 \times 0.0625 = 0.0625\) - \(P(X=1) = \binom{4}{1} (0.5)^1 (0.5)^3 = 4 \times 0.125 = 0.25\)- \(P(X=2) = \binom{4}{2} (0.5)^2 (0.5)^2 = 6 \times 0.25 = 0.375\)- \(P(X=3) = \binom{4}{3} (0.5)^3 (0.5)^1 = 4 \times 0.125 = 0.25\)- \(P(X=4) = \binom{4}{4} (0.5)^4 (0.5)^0 = 1 \times 0.0625 = 0.0625\)

Step 5: Write the Probability Mass Function (PMF)

The PMF of \(X\) is: - \(P(X=0) = 0.0625\)- \(P(X=1) = 0.25\)- \(P(X=2) = 0.375\)- \(P(X=3) = 0.25\)- \(P(X=4) = 0.0625\)

Key Concepts

Understanding the Probability Mass Function

Probability mass function (PMF) is a fundamental concept in probability theory. It describes the probabilities of the possible values of a discrete random variable. In our context, we are dealing with the random variable \(X\). This variable counts the number of heads when a fair coin is tossed four times.
The PMF provides a way to specify the probability of observing any particular outcome. We assign a probability to each potential outcome of \(X\), creating a complete distribution for \(X\).
For example, our PMF tells us that:

• The probability of no heads (\(X=0\)) is 0.0625.
• The probability of one head (\(X=1\)) is 0.25.
• The probability of two heads (\(X=2\)) is 0.375.
• ...and so on.

These probabilities must add up to 1, reflecting the certainty of observing one of the possible values of \(X\) when tossing the coin.

Clarifying a Random Variable

In probability, a random variable is a variable whose possible values are outcomes of a random phenomenon. Here, we have introduced \(X\), which represents the number of heads obtained from four coin tosses.
A random variable is a crucial concept as it translates real-world activities into mathematically manageable terms. \(X\) is considered a discrete random variable because it can take on a countable number of possible outcomes. Specifically, outcomes range from 0 to 4 heads.
To understand it better:

• \(X=0\) signifies no heads in four tosses
• \(X=1\) means one head in the sequence
• \(X=4\) corresponds to four heads (all tosses are heads)

This range of outcomes helps us apply mathematical models to determine the probability of each scenario.

Exploring the Binomial Probability Formula

The binomial probability formula is essential for calculating probabilities in scenarios involving binomial distributions. Here, we are working with a binomial distribution because we are performing repeated independent trials (coin tosses) with two possible outcomes (head or tail).
The binomial probability formula is:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where:

• \(P(X = k)\) is the probability of obtaining \(k\) successes (heads in our case).
• \(n\) is the number of trials (4 coin tosses).
• \(p\) is the probability of a single success (0.5 for a fair coin).
• \(k\) is the number of heads we're interested in.

This formula allows us to compute the probability of each possible outcome. Importantly, the combination \(\binom{n}{k}\) assists in computing the different ways \(k\) successes can occur in \(n\) trials. Using this method, all values like \(P(X=0)\) to \(P(X=4)\) can be determined.