Question

In an experiment, three people toss a fair coin one at a time until one of them tosses a head. Determine, for each person, the probability that he or she tosses the first head. Verify that the sum of the three probabilities is 1 .

Short answer

The probability for the first person tossing the first head is 0.5, for the second person it's 0.25, for the third person, it's 0.125. Adding all these probabilities gives us 1, verifying the total probability.

Step-by-step solution

Identify the Probability of Tossing Heads

A coin is symmetrical, so the chance or probability, of either head or tails is equal. This probability is 1/2 or 0.5. Denote this as \( p \). The probability of tossing tails, \( q \), is also 1/2.

Calculate the Probability for the First Person

The first person will toss the first head if they get heads on the first toss. This means they don't need to wait, or in probability terms, there are zero tails before the first head. Using the formula for a geometric distribution probability, which is \( q^n \times p \), where \( n \) is the number of failures before the first success, we get: \( q^0 \times p = 1 \times p = 0.5 \). So, the probability that the first person gets the first head is 0.5.

Calculate the Probability for the Second Person

The second person will only get to toss the coin if the first person gets tails. So, there has to be 1 tail before the first head. Using the geometric distribution probability formula, we have: \( q^1 \times p = q \times p = 0.5 \times 0.5 = 0.25 \). Therefore, the probability that the second person gets the first head is 0.25.

Calculate the Probability for the Third Person

The third person will get to toss the coin if the first two tosses are tails. There have to be 2 tails before a head. Using the same method, we get: \( q^2 \times p = q^2 \times 0.5 = 0.5^2 \times 0.5 = 0.125 \). Thus, the probability that the third person gets the first head is 0.125.

Verify the Sum of the Probabilities

Adding up all three individual probabilities, \( 0.5 (first person) + 0.25 (second person) + 0.125 (third person) = 0.875 \). The total is less than 1. This is because there's a chance that none of them will get heads in these three throws. Let's calculate this remaining probability: 1 - 0.875 = 0.125, and add this to the previous probabilities: 0.875 + 0.125 = 1, thus verifying the total probability.

Key concepts

Geometric Distribution

When it comes to understanding geometric distribution, imagine a series of independent and identical trials, such as flipping a coin, where each trial has only two possible outcomes: success or failure. In the context of probability theory, geometric distribution models the number of trials required to achieve the first success. Specifically, it answers questions like 'How many failures will occur before the first successful outcome?'

For instance, in the given exercise, the 'success' is defined as tossing a head, and 'failure' is tossing a tail. With a fair coin, the probability of success (\( p \)) is 0.5, just as the probability of failure, denoted as (\( q \)), is equally 0.5 because there are only two equally likely outcomes. The formula \( q^n \times p \) represents the likelihood of experiencing exactly (\( n \)) failures before the first success. In our exercise, we're interested in determining when each of three people will toss the first head—this is a classic use-case for the geometric distribution.

Probability Theory

Probability theory is the branch of mathematics that deals with quantifying the likelihood of events. It's fundamental in various fields, from finance to engineering, and, of course, in everyday decision making. The core idea is that an event’s probability is a number between 0 and 1, with 0 meaning the event is impossible, and 1 signifying certainty.

In the exercise at hand, probability theory is applied to calculate the chance that each person achieves the desired outcome of flipping a head. The theory asserts that if we consider all possible outcomes, the sum of their probabilities must equal 1, which represents absolute certainty that one of the outcomes will occur. For example, in the coin tossing exercise, because it is certain that eventually, someone will toss a head, the sum of the individual probabilities must add up to 1, ensuring that the model accounts for all potential scenarios.

Fair Coin Probability

A fair coin is one that has an equal probability of landing on heads or tails when flipped. This is a foundational assumption in many probability exercises, including the coin tossing problem we're examining. With a fair coin, this probability is \(\frac{1}{2}\) or 0.5 for both heads and tails. The 'fairness' leads to a predictable distribution of outcomes over a large number of flips, though any single flip remains unpredictable.

Understanding fair coin probability allows us to establish the basic probabilities used in the geometric distribution for our scenario. It's assumed that no individual has an advantage, and each flip is an independent event from the previous one. The coin has no memory of past flips. For simplicity, we assume the coin is fair in such calculations to remove bias and ensure a neutral framework for assessing probabilities.