Expectation value for first success in a binomial distribution?

In summary, the expected number of trials to get the first success in a binomial probability distribution with probability p is 1/p. This can be calculated using a clever method discussed in a forum thread.
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pellman
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This is not a homework problem. Just a curiosity. But my statistics is way rusty.

Suppose a binomial probability distribution with probability p for a success. What is the expected number of trials one would have to make to get your first success? In practice, this means if we took a large number of samples where we stopped at the first success and wrote down the number of trials N to get that success, what is the mean value of N?
 
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1. What is the definition of expectation value for first success in a binomial distribution?

The expectation value for first success in a binomial distribution is the average number of trials needed to achieve the first success in a series of Bernoulli trials, where each trial has a fixed probability of success. It is also known as the expected number of trials until the first success occurs.

2. How is the expectation value for first success in a binomial distribution calculated?

The expectation value for first success in a binomial distribution can be calculated by dividing the probability of success for each trial by the probability of failure, and then adding 1 to the result. This can also be written as (1/p) + 1, where p is the probability of success for each trial.

3. What is the significance of the expectation value for first success in a binomial distribution?

The expectation value for first success in a binomial distribution is an important measure of central tendency in the distribution. It provides an estimate of the average number of trials needed to achieve a certain outcome, and can be used to make predictions about the likelihood of achieving that outcome within a given number of trials.

4. Is the expectation value for first success in a binomial distribution the same as the mean?

No, the expectation value for first success is not the same as the mean for a binomial distribution. While both measures represent a central tendency in the distribution, the expectation value is specifically focused on the first success, while the mean takes into account all successes and failures in the distribution.

5. Can the expectation value for first success in a binomial distribution be greater than the total number of trials?

Yes, the expectation value for first success in a binomial distribution can be greater than the total number of trials. This is because the expectation value is an average and does not necessarily represent a whole number of trials. It is possible for the expectation value to be a decimal or fraction, even if the total number of trials is a whole number.

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