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 it shows heads is \(\frac{10}{11}\).

Step 1: Since there are 10 coins and they are all equally likely to be selected, the probability of selecting the fifth coin is simply 1/10. So, P(A) = 1/10. #Step 2: Find P(B|A) - Probability of the coin showing heads given it's the fifth coin#

The problem states that when the \(i^{th}\) coin is flipped, heads will appear with probability \(i/10\). Therefore, for the fifth coin (i=5), the probability of showing heads is 5/10 or 1/2. So, P(B|A) = 1/2. #Step 3: Find P(B) - Probability of heads appearing regardless of the selected coin#

Step 2: To find the probability of heads appearing, we can calculate the probability for each coin and sum them up. For each coin, we need to multiply the probability of selecting that coin (1/10) by the probability of it showing heads (i/10): \[P(B) = \sum_{i=1}^{10}{\frac{1}{10} * \frac{i}{10}} = \frac{1}{100} \sum_{i=1}^{10}{i}\] Now we can calculate the sum of the first 10 positive integers: \[\sum_{i=1}^{10}{i} = 1+2+3+\dots +10 = 55\] Therefore, P(B) = (1/100) * 55 = 11/20. #Step 4: Apply Bayes' theorem to find P(A|B) - Probability of the fifth coin given it shows heads#

Now we have all the necessary probabilities to use Bayes' theorem: \[P(A|B) = \frac{P(B|A) * P(A)}{P(B)} = \frac{(1/2) * (1/10)}{(11/20)}\] To simplify the expression, we can multiply the numerator and denominator by 20: \[P(A|B) = \frac{10}{11}\] #Solution# The conditional probability that it was the fifth coin given it shows heads is \(\frac{10}{11}\).

Key Concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory. It allows us to update the probability of a hypothesis based on new evidence. Bayes' Theorem can be written as:\[P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}\]Here, \(P(A|B)\) is the probability of event \(A\) occurring given that \(B\) has occurred. \(P(B|A)\) is the probability of event \(B\) given \(A\) is true, while \(P(A)\) and \(P(B)\) are the probabilities of \(A\) and \(B\) independently.

• Prior Probability \(P(A)\): Initial belief of hypothesis, without evidence.
• Likelihood \(P(B|A)\): The compatibility of new evidence with hypothesis.
• Evidence \(P(B)\): Overall probability of observed evidence.
• Posterior Probability \(P(A|B)\): Updated chance of hypothesis given new evidence.

In the coin example from the exercise, Bayes' theorem helps calculate the probability that a specific coin was flipped, given that it resulted in heads. This method provides a powerful tool for revising hypotheses in light of new data.

Probability Theory

Probability Theory forms the backbone for understanding how likely events are to occur. It involves the study of random events and their likelihood of occurring. The essential components in probability theory include sample spaces, events, and probabilities assigned to events.

• Sample Space: The set of all possible outcomes of a random experiment.
• Event: Any specific outcome or group of outcomes within the sample space.
• Probability: A measure ranging from 0 to 1, indicating how likely an event is to happen.

In the exercise, a set of 10 coins represents the sample space, and each potential outcome (i.e., whether it shows heads) is a specific event within this space. Probability theory helps us quantify the chances of selecting a certain coin and that coin resulting in heads.

Random Variables

Random variables are fundamental elements in the framework of probability theory. They represent numerical values that are determined by the outcome of random phenomena. There are two main types of random variables:

• Discrete Random Variables: Can take on a finite number of states (e.g., the result of rolling a die).
• Continuous Random Variables: Can take any value within a range (e.g., the exact time it will rain tomorrow).

In our exercise, the outcome of flipping one of the coins can be considered as a random variable, where each coin has a distinct probability of landing on heads.
Understanding random variables helps in defining distributions and calculating probabilities such as those needed in the given problem, where each element of a finite set (the coins and their probabilities) contributes to the overall probability of an event.

Mathematical Statistics

Mathematical Statistics is an application of probability theory. It involves analyzing data to infer properties of underlying probability distributions. This branch of mathematics provides methods for designing experiments, collecting data, and then interpreting that data statistically.
When dealing with probabilistic events, such as the coins in the exercise, statistical methods allow us to make inferences based on observed data. In cases where we seek probabilities conditional on certain observed outcomes, like in the exercise, statistics can guide the calculation, interpretation, and practical application of such probabilities.
By using mathematical statistics, we can not only find the probability of flipping that results in heads, but also gain insights into expected outcomes, variability among different coins, and test hypotheses about our observed data set. This makes mathematical statistics a crucial tool in drawing meaningful conclusions from random phenomena.