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Probability is a fundamental concept in mathematics that measures the likelihood of an event occurring. It is quantified as a number between 0 and 1 (inclusive), where 0 indicates impossibility and 1 indicates certainty. The probability of an event is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

In our exercise, the event of interest is the flip of a fair coin. Since a coin has two sides, there are two possible outcomes: heads or tails. Assuming the coin is fair, each outcome has an equal chance of occurring, so the probability of getting heads is \(\frac{1}{2}\) or 0.5. This means that if you flip a fair coin once, you have a 50% chance of getting heads.

The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, with a constant probability of success in each trial. To qualify as a binomial experiment, the process must have a fixed number of trials, two possible outcomes (success or failure), and the probability of success must be constant in each trial.

In our case, flipping a fair coin 100 times is a perfect example of a binomial experiment. We have a fixed number of trials (100 coin tosses), two possible outcomes (heads for success, tails for failure), and the probability of getting heads (success) is constant at 0.5 for each flip. By using the binomial distribution, we can predict how many times heads will come up in those 100 flips. The expected number of heads can thus be calculated by multiplying the number of trials (100) by the probability of success (0.5), resulting in 50 expected heads.

A fair coin toss is one where the coin is unbiased and has no preference for landing on heads or tails, making the probability for either side landing up equal. When flipping a fair coin, the expectation for the number of heads or tails is based on the inherent fairness, which inversely provides us with a handy way to test if a coin is indeed fair.

The concept of expected value is a type of average used in probability, representing the anticipated value of a given random variable after many repetitions of an experiment. It is found by multiplying each possible outcome by their respective probabilities and adding these products together. In our exercise with the fair coin, since the probability is 0.5 for heads and the number of trials is 100, the expected value calculation matches our intuitive estimate: \(\text{Expected value} = 100 \times 0.5 = 50\), where 50 is the number of heads you would expect to see after 100 fair coin tosses.