Problem 45 - A First Course in Probability

Question

Suppose we have 10 coins such that if the \(i\) th coin is flipped, heads will appear with probability \(i / 10, i=\) \(1,2, \ldots, 10 .\) When one of the coins is randomly selected and flipped, it shows heads. What is the conditional probability that it was the fifth coin?

Step-by-Step Solution

Answer

The conditional probability that it was the fifth coin, given that it showed heads, is \(\frac{1}{11}\).

Step 1: Find the unconditional probability of choosing the 5th coin

Using the information given, there are 10 coins, and we randomly select one coin, so the probability of choosing the 5th coin is 1/10.

Step 2: Calculate the probability of flipping heads given that the 5th coin is selected

According to the problem, if the i-th coin is flipped, heads will appear with probability i/10, so for the 5th coin, the probability of flipping heads is 5/10 or 1/2.

Step 3: Calculate the unconditional probability of flipping heads

Since each coin has its own specific probability of showing heads, we have to sum up the probabilities for each coin multiplied by the probability of choosing each coin. Remember that the probability of choosing any of the 10 coins is 1/10. Thus, the unconditional probability of flipping heads is given by the sum: P(Heads) = Sum(P(Heads | coin i) * P(coin i)) for i = 1 to 10, P(Heads) = (1/10)(1/10) + (2/10)(1/10) + ... + (10/10)(1/10) = (1/10) * Sum(i/10), for i = 1 to 10. To find the sum of i/10 for i = 1 to 10, we can write it as: Sum(i/10) = (1+2+3+4+5+6+7+8+9+10)/10 = 55/10 So, P(Heads) = (1/10) * (55/10) = 11/20

Step 4: Use Bayes' theorem to find the conditional probability

Bayes' theorem states that the conditional probability is given by: P(5th coin | Heads) = P(Heads | 5th coin) * P(5th coin) / P(Heads) Using the values calculated in Steps 1, 2, and 3: P(5th coin | Heads) = (1/2) * (1/10) / (11/20) To simplify this expression, first, we can multiply both numerator and denominator by 20: P(5th coin | Heads) = (20/2) * (1/10) / 11 Now, dividing 20/2 gives 10: P(5th coin | Heads) = 10 * (1/10) / 11 Cancel out the 10s: P(5th coin | Heads) = 1/11 So the conditional probability that it was the 5th coin, given that it showed heads, is 1/11.

Key Concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory that helps us find the probability of an event based on prior knowledge of related events. In this exercise, we want to determine the probability that the fifth coin was chosen given that the result of a flip showed heads. Using Bayes' Theorem involves several key probabilities:

• P(Heads | 5th coin): The probability of flipping a head if we flipped the 5th coin, which is \( \frac{1}{2} \).
• P(5th coin): This is the probability of choosing the 5th coin initially, \( \frac{1}{10} \), because every coin has an equal chance of being selected.
• P(Heads): The unconditional probability of flipping heads, considering all coins, which was calculated as \( \frac{11}{20} \).

Plugging these into Bayes' Theorem gives us:\[ P(\text{5th coin | Heads}) = \frac{P(\text{Heads | 5th coin}) \times P(\text{5th coin})}{P(\text{Heads})}\]Finally, we calculated that the probability is \( \frac{1}{11} \). Bayes' Theorem helps us update our beliefs about which coin was flipped based on the outcome of the flip.

Unconditional Probability

Unconditional probability, sometimes known as marginal probability, is the likelihood of an event happening regardless of whether any related events occur. It's simply the overall probability of a scenario without conditioning on any other factors.In the solution process, this concept comes in when calculating the probability of flipping heads without considering which particular coin was selected. To find this, we sum the probabilities of flipping heads for each coin, each weighted by the probability of choosing that coin.Here's the step to find the unconditional probability of flipping heads, denoted by P(Heads):

• Each coin can be chosen with a probability of \( \frac{1}{10} \), since there are ten coins.
• The probability of heads for each coin depends on its index, such as \( \frac{1}{10}, \frac{2}{10}, ..., \frac{10}{10} \).
• Therefore, the probability P(Heads) is calculated by \( \frac{1}{10} \times \frac{1}{10} + \frac{1}{10} \times \frac{2}{10} + \ldots + \frac{1}{10} \times \frac{10}{10} \), resulting in a total of \( \frac{11}{20} \).

This represents the average chance of obtaining heads with any coin, setting the stage for subsequent Bayesian calculations.

Probability Theory

Probability theory is the branch of mathematics that deals with uncertainty and random events. It provides a framework for reasoning about randomness and helps quantify how likely an event is to happen. In our exercise, we make use of key principles from this theory to solve a real-world problem about randomly flipping a coin.Key elements of probability theory utilized here include:

• Random Selection: We have a set of 10 coins, each equally likely to be selected for flipping.
• Event Probability: Each coin has a specific probability \( i/10 \) of landing heads, where \( i \) represents the coin's position or index.
• Summation of Probabilities: We calculated the overall likelihood of flipping a head by considering each coin's contribution.

Understanding these concepts enabled us to navigate through the calculations of unconditional and conditional probabilities in the example. Probability theory provides the guidelines and mathematical tools necessary to manage situations involving chance.