Problem 143 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Assertion: A bag contains \(n+1\) coins. It is known that one of these coins has a head on both sides while the other coins are fair. One coin is selected at random and tossed. If head turns up, the probability that the selected coin was fair, is \(\frac{n}{n+2}\) Reason: If an event \(A\) occurs with two mutually exclusive and exhanstive events \(E_{1}\) and \(E_{2}\), then \(P(E / A)\) \(=\frac{P\left(E_{i}\right) P\left(A / E_{i}\right)}{P\left(E_{1}\right) P\left(A / E_{1}\right)+P\left(E_{2}\right) P\left(A / E_{2}\right)}, i=1,2 .\)

Step-by-Step Solution

Answer

The probability is \( \frac{n}{n+2} \).

Step 1: Understanding the Problem

We need to find the probability that the selected coin is a fair coin given that when tossed, a head shows up. In probability terms, we are looking for \( P(F\mid H) \), where \( F \) is the event that a fair coin is selected and \( H \) is the event that the coin toss resulted in a head.

Step 2: Identifying Given Events

Let \( E_1 \) be the event of selecting the fair coin and \( E_2 \) be the event of selecting the double-headed coin. Since there are \( n+1 \) coins (including one double-headed coin), \( P(E_2) = \frac{1}{n+1} \) and \( P(E_1) = \frac{n}{n+1} \) because the rest \( n \) coins are fair.

Step 3: Analyzing Outcomes of Tossing

For the double-headed coin (event \( E_2 \)), \( P(H \mid E_2) = 1 \) because both sides are heads. For a fair coin (event \( E_1 \)), \( P(H \mid E_1) = \frac{1}{2} \) because there is an equal chance of head or tail.

Step 4: Applying Bayes' Theorem

Using Bayes' theorem, \( P(F \mid H) = \frac{P(E_1) P(H \mid E_1)}{P(E_1) P(H \mid E_1) + P(E_2) P(H \mid E_2)} \). Substituting the values: \[ P(F \mid H) = \frac{\frac{n}{n+1} \cdot \frac{1}{2}}{\frac{n}{n+1} \cdot \frac{1}{2} + \frac{1}{n+1} \cdot 1} \] Simplifying gives: \[ P(F \mid H) = \frac{n}{n+2} \].

Step 5: Comparing with Given Assertion

The calculated probability \( \frac{n}{n+2} \) matches the assertion in the problem. Thus, the assertion that the probability of having a fair coin given a head appears is indeed \( \frac{n}{n+2} \).

Key Concepts

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability and statistics, used to update the probability estimate for an event as more information becomes available. It is a way to figure out how likely an event is based on prior knowledge of conditions that might be related to the event. In essence, it allows us to calculate the probability of an event, given the probability of another related event has occurred.
Here is the formula:
\[P(A | B) = \frac{P(B | A) \cdot P(A)}{P(B)}\]- \(P(A | B)\) is the probability of event \(A\) given event \(B\) has occurred.- \(P(B | A)\) is the probability of event \(B\) given event \(A\) is true.- \(P(A)\) and \(P(B)\) are the probabilities of observing \(A\) and \(B\) independently of each other.
In the original exercise, Bayes' theorem helps us calculate the probability that the selected coin was fair, given that we observed a head turn up when tossed. This involves inputs from both the fair and double-headed coin scenarios, updating our previous assumptions based on new evidence.

Mutually Exclusive Events

In probability theory, mutually exclusive events are events that cannot occur simultaneously. If one event happens, the other cannot, which makes their joint probability zero. This is a key concept because it simplifies calculations in many problems where events do not overlap.
Given events \(A\) and \(B\):

• They are mutually exclusive if \(P(A \cap B) = 0\).
• For example, in a single coin toss, getting heads (\(H\)) and tails (\(T\)) are mutually exclusive events since one cannot happen at the same time as the other.

In the exercise context, selecting a fair coin and selecting the double-headed coin are mutually exclusive because you can only select one of the coins at a time. However, unlike some scenarios, these two events are not collectively exhaustive since they don't account for all possible outcomes when other factors or coins aren't considered.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already happened. It helps to refine our predictions based on new information.
The formula for conditional probability is:
\[P(A | B) = \frac{P(A \cap B)}{P(B)}\]- \(P(A | B)\) is the probability of event \(A\) given event \(B\) has occurred.- \(P(A \cap B)\) is the probability of both events \(A\) and \(B\) occurring together.- \(P(B)\) is the probability of event \(B\) occurring.
In the original problem, conditional probability is used to determine \(P(F | H)\) – the probability that the selected coin is fair given that the toss resulted in heads. This is crucial because it distinguishes between the scenarios of tossing the selected coins and observing the outcomes. The heads result (\(H\)) impacts the chance of having started with a fair coin (\(F\)), prompting necessary probability adjustments like seen in Bayes' theorem application.

Fair Coin

A fair coin is an idealized concept in probability theory. It is a coin that has an equal chance of landing heads or tails, assuming no bias in any other form. In mathematical terms, each side of a fair coin has a probability of \(\frac{1}{2}\) of being the result when the coin is flipped.
Characteristics of a fair coin include:

• No side is heavier; thus, it's unbiased.
• It has a symmetric shape ensuring equal chance.
• The probability of heads \( (P(H) = \frac{1}{2}) \) and tails \( (P(T) = \frac{1}{2}) \) always sums up to 1.

Within the provided exercise, the fair coins and the unique double-headed coin play essential roles in determining probabilities; particularly, calculating the likelihood of the selected coin being fair if heads show on a toss. Here, understanding the nature of a fair coin is vital as it serves as the baseline for contrast against the double-headed coin's behavior.