# Expected value flip of a coin

• Somefantastik
In summary, the expected value of flipping a coin is 0.5, calculated by multiplying the probabilities of each outcome by their respective values. It is significant in understanding the likelihood of outcomes in random experiments. The expected value can be different from 0.5 if the coin is biased. The law of large numbers states that as the number of trials increases, the average result will approach the expected value, which is helpful in predicting outcomes in random experiments.

#### Somefantastik

[problem] Consider n independent flips of a coin having prob p of landing heads. Say a changeover occurs whenever an outcome differs from the one preceding it. E.g., H H T H T H H T means 5 changeovers. What is the expected number of changeovers for an arbitrary p?

[s0lution]
see attachment.

WHERE is the exponential coming from?

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Thank you! We haven't gotten there in class yet, so I was clueless. Thanks again.

## 1. What is the expected value of flipping a coin?

The expected value of flipping a coin is 0.5. This means that if you were to flip a coin an infinite number of times, the average result would be 0.5 or 50% heads and 50% tails.

## 2. How is the expected value of flipping a coin calculated?

The expected value of flipping a coin is calculated by taking the probability of each possible outcome (heads or tails) and multiplying it by the value of that outcome (1 for heads and 0 for tails). In the case of a fair coin, both outcomes have equal probabilities, so the expected value is (0.5 x 1) + (0.5 x 0) = 0.5.

## 3. What is the significance of the expected value in flipping a coin?

The expected value in flipping a coin is significant because it represents the average outcome of an experiment that has random and uncertain results. It allows us to make predictions and decisions based on probabilities and helps us understand the likelihood of certain outcomes.

## 4. Can the expected value of flipping a coin be different from 0.5?

Yes, the expected value of flipping a coin can be different from 0.5 if the coin is biased. A biased coin is one that has a higher probability of landing on one side compared to the other. In this case, the expected value will be different and will depend on the specific probabilities of each outcome.

## 5. How does the law of large numbers relate to the expected value of flipping a coin?

The law of large numbers states that as the number of trials in an experiment increases, the average of the results will approach the expected value. In the case of flipping a coin, if you were to flip it an infinite number of times, the average result would approach the expected value of 0.5. This law helps us understand the predictability of outcomes in random experiments.