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: Understand the Scenario

We have two coins in the bag. One coin is fair (with heads and tails) and the other is a two-headed coin. We randomly pick one coin and flip it, observing a heads.

Step 2: Define the Events

Let A be the event of picking the fair coin and H be the event of getting heads. We are tasked to find the probability of A given H, denoted as \( P(A|H) \).

Step 3: Use Bayes' Theorem

Bayes' theorem gives us \( P(A|H) = \frac{P(H|A)P(A)}{P(H)} \). We now need to find each of these probabilities.

Step 4: Determine P(A), P(H|A), and P(H)

The probability of picking the fair coin, \( P(A) \), is \( \frac{1}{2} \), as each coin is equally likely to be picked. \( P(H|A) \), the probability of getting heads with the fair coin, is also \( \frac{1}{2} \). \( P(H) \), the probability of getting heads overall, is calculated using \( P(H) = P(H|A)P(A) + P(H|B)P(B) \), where \( P(B) \) is the probability of picking the two-headed coin. Because the two-headed coin always gives heads, \( P(H|B) = 1 \) and \( P(B) = \frac{1}{2} \).

Step 5: Calculate P(H)

Substituting into the equation above, \( P(H) = \frac{1}{2} \times \frac{1}{2} + 1 \times \frac{1}{2} = \frac{1}{4} + \frac{1}{2} = \frac{3}{4} \).

Step 6: Calculate P(A|H)

Now we use Bayes' Theorem to calculate \( P(A|H) = \frac{P(H|A)P(A)}{P(H)} = \frac{\frac{1}{2} \times \frac{1}{2}}{\frac{3}{4}} = \frac{1}{4} \div \frac{3}{4} = \frac{1}{3} \).

Key Concepts

Understanding Probability

Probability is a measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1, where 0 means the event will not occur, and 1 means the event will certainly occur. In a mathematical sense, probability helps us quantify uncertainty. In the context of our exercise, when you pick a coin and flip it, you are dealing with probability. Here, you have two equally likely scenarios since there are two coins: either you pick the fair coin or the biased coin which always shows heads. Since both events are equally likely, each one has a probability of 0.5 or 50%. This is a fundamental idea that sets the stage for solving more complex problems involving probability.

Unpacking Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted by \( P(A|B) \), which reads as the probability of event A given that event B has occurred.Bayes' Theorem is a powerful formula in probability which allows us to update our predictions or hypotheses with new evidence. It expresses the conditional probability of an event, based on prior knowledge of conditions related to the event.

• For our problem, we wanted to find the probability that the coin is fair, given that we observed heads.
• This required us to use prior probabilities: the probability of picking each coin and the probability of each event occurring.

Bayes' Theorem calculates \( P(A|H) \) as:\[ P(A|H) = \frac{P(H|A)P(A)}{P(H)} \]This formula helps us understand the dependence between different events.

Fair Coin vs Biased Coin

Coins can be interesting subjects of probability studies because they represent events with clear, binary outcomes: heads or tails. A fair coin has two sides, heads and tails, giving an equal chance of landing on heads or tails each time you flip it - thus each side has a probability of \( \frac{1}{2} \).On the other hand, a biased coin in this scenario has two heads, meaning it will always land on heads, and hence has a probability of 1 for heads and 0 for tails. This distinct difference is crucial when analyzing the results of a coin toss involving both a fair and a biased coin.Understanding these differences allows us to calculate probabilities about which coin was picked based on the result of a coin flip. Here, we use the knowledge that seeing heads could have resulted from picking either coin, but the probability is different based on which coin you choose. This further ties into Bayes' Theorem, allowing us to backtrack from the observed result, heads, to determine the likelihood of having chosen the fair coin.