Animated Solution for Mathematics - Probability: A fair coin is tossed repeatedly. If the tail appears on first four tosses, then the probability of the head appearing on the fifth toss equals

Why Your Intuition Might Be Failing You

Imagine you are standing in a casino, watching a roulette wheel or a simple coin toss. You see four tails in a row. Your brain, wired to seek patterns and balance, screams that a head must be 'due' to appear.

This is the human condition—we are pattern-seeking machines. But in the cold, objective world of mathematics, this intuition is the source of a famous trap known as the Gambler's Fallacy. Today, we are going to dismantle this misconception by looking at the beautiful, simple logic of independent events.

Visualizing the Experiment

Let us start by visualizing our experiment. We are tossing a fair coin five times. Imagine five empty slots on a table, each waiting to be filled by either a Head ($H$) or a Tail ($T$).

The problem states that the first four slots are already filled with Tails: $T_1,T_2,T_3,T_4$. We are now standing at the fifth slot, the final frontier of our experiment, and we want to know the probability of a Head appearing there.

Our objective is to calculate the conditional probability:

Independence

Here is the pivot point of our entire journey. What does it mean for a coin to be 'fair'? It means it is unbiased, and for any single toss, the sample space is $S=\{H,T\}$, with:

Crucially, a coin has no memory. It does not 'know' that it just landed on tails four times. It does not 'feel' the need to balance the scales.

Because the outcome of the fifth toss is completely unaffected by the previous four, we classify these as Independent Events. In probability theory, if two events $A$ and $B$ are independent, the conditional probability $P(A|B)$ is simply $P(A)$.

The condition $B$ provides no information that changes the likelihood of $A$. Mathematically, this means:

The information about the first four tails is, in the language of probability, redundant.

Trusting the Math

Once we strip away the psychological noise of the 'streak,' the problem collapses into the simplest form imaginable. We are just looking for the probability of a head on a single, fair toss.

We know that:

The previous four tails do not change this value by even a fraction of a percent. The math is elegant, clean, and completely indifferent to our human desire for patterns. The final answer is $\frac{1}{2}$, which corresponds to option (a).

A Lesson for JEE

As you prepare for your exams, remember this: JEE problems often include 'distractor' information designed to test your conceptual clarity. They want to see if you will get bogged down in unnecessary calculations or if you can identify the core principle—in this case, the independence of events.

Whenever you see a sequence of events, pause and ask yourself: 'Does the past influence the future here?' If the answer is no, you have found an independent system. Trust the math, ignore the noise, and you will find the path to the correct answer every single time.