Problem 33 - Calculus for Biology and Medicine

Question

Toss a fair coin 10 times. Let \(X\) be the number of heads. Find (a) \(P(X=5)\). (b) \(P(X \geq 8)\). (c) \(P(X \leq 9)\).

Step-by-Step Solution

Answer

(a) \(P(X=5) \approx 0.2461\) (b) \(P(X \geq 8) \approx 0.0547\) (c) \(P(X \leq 9) \approx 0.9990\)

Step 1: Understanding the Problem

This exercise involves binomial probability, as we are tossing a fair coin multiple times and counting the number of successful outcomes (heads). Since the coin is fair, the probability of getting a head in a single toss is \(p = 0.5\). We need to calculate the probabilities associated with specific outcomes using the binomial probability formula.

Step 2: Identify the Binomial Distribution

We toss the coin 10 times, making this a binomial experiment with \( n = 10\) and \( p = 0.5\). Each outcome (head or tail) in each toss is independent. The random variable \(X\), representing the number of heads, follows a binomial distribution: \(X \sim \text{Binomial}(10, 0.5)\).

Step 3: Recall the Binomial Probability Formula

The probability that you get exactly \(k\) heads in \(n\) tosses of a fair coin is given by the formula:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(\binom{n}{k}\) is the binomial coefficient, which calculates 'n choose k' combinations.

Step 4: Solve Part (a) - Find \(P(X=5)\)

To calculate \(P(X=5)\), use the binomial formula:\[P(X = 5) = \binom{10}{5} (0.5)^5 (1-0.5)^{10-5} = \binom{10}{5} (0.5)^{10}\]Compute \(\binom{10}{5} = 252\). Thus:\[P(X = 5) = 252 \times 0.5^{10} = 252 \times \frac{1}{1024} \approx 0.2461\]

Step 5: Solve Part (b) - Find \(P(X \geq 8)\)

To find \(P(X \geq 8)\), calculate \(P(X=8)\), \(P(X=9)\), and \(P(X=10)\), then sum the probabilities:\[ P(X = 8) = \binom{10}{8} (0.5)^{8} (0.5)^{2} \ P(X = 9) = \binom{10}{9} (0.5)^{9} (0.5)^{1} \ P(X = 10) = \binom{10}{10} (0.5)^{10} \]Compute these values:\(\binom{10}{8} = 45\), \(\binom{10}{9} = 10\), and \(\binom{10}{10} = 1\).\[P(X \geq 8) = 45 \times 0.5^{10} + 10 \times 0.5^{10} + 1 \times 0.5^{10} = 56 \times \frac{1}{1024} = \frac{56}{1024} \approx 0.0547\]

Step 6: Solve Part (c) - Find \(P(X \leq 9)\)

Since all possibilities sum to 1, calculate \(P(X \leq 9)\) by noting:\[ P(X \leq 9) = 1 - P(X = 10) \]We previously computed \(P(X = 10) = \frac{1}{1024}\), so:\[ P(X \leq 9) = 1 - \frac{1}{1024} = \frac{1023}{1024} \approx 0.9990\]

Key Concepts

Binomial Probability Formula

Binomial distribution is utilized in scenarios where there are fixed numbers of trials, each with two possible outcomes: success or failure. In our exercise, tossing a fair coin represents this concept, as each toss results in either a head (success) or a tail (failure). The **binomial probability formula** is the mathematical expression that helps determine the probability of obtaining exactly \(k\) successes in \(n\) independent trials. It is represented as follows:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]
Here:

• \(\binom{n}{k}\) is the **binomial coefficient**—explained further below—that computes the number of combinations for selecting \(k\) successes out of \(n\) trials.
• \(p^k\) suggests the probability of achieving success in exactly \(k\) of the trials, where \(p\) is the probability of success on a single trial.
• \((1-p)^{n-k}\) represents the probability of obtaining failures in the remaining trials.

The formula allows for calculating specific outcomes like getting exactly 5 heads in 10 coin tosses, providing a practical tool for predicting probabilities when dealing with binomial distributions.

Random Variable

In probability and statistics, a **random variable** is a numerical description of the outcome of a statistical experiment. For our exercise, the random variable \(X\) is used to represent the number of heads obtained when tossing a fair coin 10 times. Random variables can take on different values based on the experiment's outcome.

• A random variable can be either discrete or continuous. Here, \(X\) is discrete because it can only take on integer values (0, 1, 2,...,10) based on the number of successful trials (heads).
• The actual value of \(X\) is determined by the experiment. For example, if you get 5 heads in the 10 coin tosses, then \(X = 5\).
• The random variable \(X\) follows a binomial distribution in this scenario, since there are fixed trial numbers with only two possible outcomes per trial.

This conceptual understanding of a random variable is essential for applying mathematical models in probability and helps in predicting outcomes based on given conditions like in binomial distributions.

Binomial Coefficient

The **binomial coefficient** is a crucial component of the binomial probability formula. It is used to compute the number of ways \(k\) successes can occur in \(n\) trials and is denoted as \(\binom{n}{k}\).

The binomial coefficient is calculated using the formula:
\[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \]

• Here, \(n!\) (n factorial) refers to the product of all positive integers up to \(n\).
• Similarly, \(k!\) and \((n-k)!\) are the factorials of \(k\) and \((n-k)\) respectively.

This calculation gives us the number of possible ways to achieve exactly \(k\) successes in \(n\) trials. For example, in our problem, \(\binom{10}{5} \) computes how many different ways we can get 5 heads out of 10 tosses, which is 252. Understanding this allows you to deconstruct problems involving combinations of outcomes in experiments like the coin toss exercise.