You and a friend are talking about the probability of getting a heads on a single toss of a fair coin. Your friend insists that you are more likely to get a heads on a single toss of a fair coin than a tails. Is your friend correct, why or why not? If we were to toss the fair coin an infinite number of times, what would we expect?

In order to address the question of whether it is more likely to obtain a heads or a tails on a single toss of a fair coin, let us first delve into the concept of probability. Probability, in mathematics, measures the likelihood of an event occurring. In the case of a fair coin, where both heads and tails are equally likely outcomes, the probability of each outcome is 1/2 or 50%.

Now, your friend insists that getting a heads is more likely than getting a tails on a single toss of a fair coin. However, this claim is incorrect. The probability of getting a heads and the probability of getting a tails on a single toss of a fair coin are both equal, namely 1/2 or 50%. Each toss of the coin is an independent event, meaning the outcome of previous tosses does not influence the outcome of the current toss. Therefore, the likelihood of obtaining a heads or a tails remains the same regardless of previous results.

To illustrate this further, consider a hypothetical experiment where you toss the fair coin 100 times and record the outcomes. If the coin is truly fair, you would expect to see approximately 50 heads and 50 tails. However, due to random variation, the actual number of heads and tails may deviate slightly from the expected value of 50. Nevertheless, as the number of tosses increases, the discrepancy between the observed and expected number of heads and tails diminishes. This phenomenon is one of the fundamental concepts in probability theory known as the law of large numbers.

Now, let us contemplate what would happen if we were to toss the fair coin an infinite number of times. In this hypothetical scenario, the law of large numbers tells us that the observed frequency of obtaining heads or tails would approach their respective probabilities of 1/2 or 50%. This means that, on average, half of the outcomes would be heads and the other half would be tails. However, it is important to note that even after an infinite number of tosses, there is still a possibility of obtaining a sequence of all heads or all tails, as long as the events are independent.

In the context of infinite tosses, the concept of expected value is also relevant. The expected value, often denoted as E(X), represents the average or long-term outcome of a random variable. For each toss of the fair coin, the random variable can take on two values, heads (H) or tails (T), each with a probability of 1/2. In this case, the expected value of the random variable X can be calculated as:

E(X) = (1/2 * H) + (1/2 * T) = (1/2 * 1) + (1/2 * 0) = 1/2.

Therefore, if we were to toss the fair coin an infinite number of times, we would expect that the long-term average outcome would be 1/2, indicating that half of the tosses would result in heads and the other half in tails.

In conclusion, your friend’s claim that obtaining a heads is more likely than obtaining a tails on a single toss of a fair coin is incorrect. Each outcome of a fair coin toss has an equal probability of occurring, namely 1/2 or 50%. Throughout numerous tosses, the observed frequency of heads and tails will converge to their respective probabilities. In the case of an infinite number of tosses, the long-term average outcome would be 1/2, indicating an equal likelihood of heads and tails.