Problem 39 - Calculus for Biology and Medicine

Question

A bag contains two coins, one fair and the other with two heads. You pick one coin at random and flip it. What is the probability that you picked the fair coin given that the outcome of the toss was heads?

Step-by-Step Solution

Answer

The probability is \( \frac{1}{3} \).

Step 1: Define the events

Let \( F \) be the event of picking the fair coin and \( H \) be the event of getting heads. We want to find \( P(F \mid H) \), the probability that we picked the fair coin given that we got heads.

Step 2: Apply Bayes' Theorem

According to Bayes' Theorem, \( P(F \mid H) = \frac{P(H \mid F) \cdot P(F)}{P(H)} \). We need to calculate each part of this formula.

Step 3: Calculate \( P(F) \)

The probability of picking the fair coin, \( P(F) \), is \( \frac{1}{2} \) since there are two coins and one of them is fair.

Step 4: Calculate \( P(H \mid F) \)

The probability of flipping heads given that we've picked the fair coin is \( \frac{1}{2} \), since it has one head and one tail.

Step 5: Calculate \( P(H) \)

To find \( P(H) \), consider all the ways to get heads: either by picking the two-headed coin which always results in heads or by picking the fair coin and getting heads. Thus, \( P(H) = P(H \mid F) \cdot P(F) + P(H \mid \text{Two-heads}) \cdot P(\text{Two-heads}) = \frac{1}{2} \cdot \frac{1}{2} + 1 \cdot \frac{1}{2} = \frac{1}{4} + \frac{1}{2} = \frac{3}{4} \).

Step 6: Compute \( P(F \mid H) \)

Now plug the values into Bayes' Theorem: \( P(F \mid H) = \frac{P(H \mid F) \cdot P(F)}{P(H)} = \frac{ \frac{1}{2} \cdot \frac{1}{2} }{ \frac{3}{4} } = \frac{1}{4} \cdot \frac{4}{3} = \frac{1}{3} \). Thus the probability of picking the fair coin given that you flipped a head is \( \frac{1}{3} \).

Key Concepts

Conditional Probability

Conditional probability is an essential concept in statistics and probability theory. It describes the likelihood of an event occurring given that another event has already occurred. In our exercise, we want to find the probability that we selected the fair coin, given that the coin flip resulted in heads.

Bayes' Theorem is a powerful tool for solving conditional probability problems. It allows us to update our prediction about an event when given new evidence. Specifically, it helps us find the conditional probability by using the formula:\[P(F \mid H) = \frac{P(H \mid F) \cdot P(F)}{P(H)}\]Where:

• \(P(F \mid H)\) is the probability of selecting the fair coin after observing a head.
• \(P(H \mid F)\) is the likelihood of a head when flipping the fair coin.
• \(P(F)\) is the probability of initially picking the fair coin from the bag.
• \(P(H)\) is the probability of getting heads with any chosen coin.

This powerful theorem bridges the gap between evidence and belief, enhancing our understanding of probabilities in uncertain situations.

Fair and Biased Coins

Coins that are fair have an equal chance of landing on heads or tails. In mathematical terms, this means that each side has a probability of \( \frac{1}{2} \). In contrast, a biased coin doesn't follow this rule. In our exercise, the biased coin is a two-headed coin, i.e., every flip results in heads, giving heads a probability of 1 and tails a probability of 0.

Understanding fair and biased coins helps us analyze different outcomes in probability exercises. For our situation:

• The fair coin behaves normally with a 50% chance for each side.
• The biased coin only shows heads with a probability of 1.

Differentiating between these two types of coins is crucial, as it affects the calculation of likelihoods in probability theory and impacts our understanding of realistic scenarios.

Probability Theory

Probability theory is the branch of mathematics that deals with quantifying uncertainty. It provides a framework for predicting the likelihood of different outcomes. Key concepts in probability theory include events, probability of events, and the relationships between these probabilities.

In our exercise, we'd applied fundamental principles of probability theory to determine the likelihood of choosing the fair coin when a head is observed. By combining conditional probability with our understanding of the different coin types, we used:

• Basic calculations, such as the probability of choosing each coin.
• Calibrating each step to include all potential outcomes when getting a head.

This rational approach enables us to understand and solve complex probability problems, even ones involving seemingly simple objects like coins. With probability theory, we can make informed predictions and validate assumptions based on numerical data.