Six programmers from Alfa Romalfa decide to toss coins on an...

Six programmers from Alfa Romalfa decide to toss coins on an "odd person out" basis to determine who will buy the coffee. Thus, there will be a loser if exactly one of the coins falls heads or exactly one falls tails. ${ }^7$ What is the probability that the outcome will be decided on the first toss? What is the probability that exactly four trials will be required? Not more than four?

Key Concepts

Complement Rule

The complement rule is a fundamental principle in probability that states the probability of an event not occurring is one minus the probability that it does occur. In the scenario where the problem asks for the probability that a decision is reached not more than four trials, this rule helps in deriving probabilities by subtracting the unwanted outcomes from the total probability space.

Independent Trials

The coin tosses in the problem are independent events, meaning the result of one coin toss does not affect the outcome of another. This property allows the probability of a sequence of outcomes to be determined by multiplying the probabilities of individual outcomes, which is essential when calculating the probability of a particular pattern across multiple trials.

Geometric Distribution

The geometric distribution is relevant when determining the probability that the first successful occurrence of an event happens on a specific trial. In the context of the question, once the probability of a ‘decision’ in a single round is known, the geometric distribution helps in calculating the probability that exactly four trials are needed for the first occurrence of a decision outcome.

Binomial Distribution

This concept applies to experiments that consist of a fixed number of independent trials, each with two possible outcomes (such as heads or tails), where the probability of success remains constant. In this question, the coin tosses are modeled using a binomial distribution to calculate the probability of obtaining exactly one head or exactly one tail among multiple tosses.

Combinatorial Analysis

Combinatorial analysis is used to count the number of ways an event can occur without having to enumerate every possibility. Here, binomial coefficients are used to determine the number of combinations in which a specific outcome (exactly one head or one tail) can be achieved among a fixed number of coin tosses.

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