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Probability is a fundamental concept used to measure the likelihood of a particular event occurring. It's the backbone of statistical analysis and informs decisions in a wide range of fields, from science to finance. In essence, if we say that there's a probability of 0.5 for flipping heads on a coin, we mean that there is a 50% chance that this will be the outcome. Probability values range from 0 to 1, where 0 means an event is impossible, and 1 means it is certain to occur.

When flipping a fair coin, we expect heads and tails to be equally likely. This is expressed by probabilities of 0.5 for both outcomes, and over many flips, we expect to see roughly as many heads as tails. However, probabilities only tell us the expected likelihood in the long run and not the certainty of outcomes in the short term. For example, it is still possible, although unlikely, to flip heads several times in a row.

Independent events are pivotal in understanding why probabilities operate the way they do. Two events are considered independent if the occurrence of one does not affect the probability of the other occurring.

Keeping the coin flip example in mind, each flip is an independent event. The coin doesn't 'remember' what happened on the previous flip; thus, the probability of getting heads remains 0.5 on every single flip, regardless of past outcomes. It's a common misconception to think that if a coin has landed on tails several times in a row, it is 'due' to land on heads. This is known as the gambler's fallacy. In reality, each flip is a fresh event with the same 0.5 probability for heads and 0.5 for tails. This element of independence is crucial for the law of large numbers to hold true.

The expected value is the long-term average outcome of a random event if it is repeated many times. For a fair coin flip, the expected value for the number of heads is 0.5 per flip, since you're equally likely to get heads or tails.

The law of large numbers is a principle that says as you perform the same random experiment (like flipping a coin) over and over, the average of the results will get closer to the expected value. In our coin-flipping scenario, as the number of flips gets very large, the proportion of heads is expected to edge closer to 0.5, meaning that for a large number of flips, approximately half should be heads.