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In probability, independent events are events where the outcome of one event does not affect the outcome of another. For instance, when you toss a coin, the result of a toss—whether it lands on heads or tails—does not influence what will happen on the next toss.

Each coin flip is its own separate event. This means even if you toss a coin five times and get the sequence "TITTT", the next flip is still a completely new and independent event.

It can still equally be heads or tails in the next flip, regardless of the number of previous tails or heads.

A fair coin toss means that the coin is unbiased. Both sides of the coin, heads and tails, have an equal chance of appearing when the coin is flipped.

In mathematical terms, a fair coin toss is symmetric. This means the probability for each side, heads or tails, is the same. There are no external factors influencing the result, ensuring it's completely random each time you toss it.

Since both outcomes are equally likely, this fairness underscores the property of independent events in coin tosses.

Calculating probability is a fundamental concept in determining how likely it is for an event to occur. For a fair coin, which has two sides, the probability of the coin landing on either side (heads or tails) is calculated as follows:

- Probability of heads = \( P(H) = \frac{1}{2} \)
- Probability of tails = \( P(T) = \frac{1}{2} \)

The formula for calculating probability ensures that the total probabilities of all possible outcomes add up to 1. Each toss of the coin reaffirms this balance, reflecting that previous outcomes do not alter the prospects of any upcoming toss.

Equal likelihood is a principle highlighting that, for certain random processes, the possible outcomes are all equally probable. In the scenario of a fair coin toss, the terms 'equal likelihood' describe the fundamental nature where heads and tails each have a 50% chance of landing face up.

This concept is crucial for understanding why, in a series of coin tosses, each flip remains balanced in terms of probability. No string of previous results, such as a sequence of tails, skews or adjusts the likelihood of the next result. The inherent fairness of the coin's design ensures that every toss remains equally likely to result in heads or tails, making predictions unbiased and equitable across trials.