# Probability: Historic Probability and Expected Value

Leaders of a local club want to focus on recruiting and looked through their recruiting and membership paperwork. They realized that over the past 30 years of annual open-house events, they spoke to four people who were interested enough in joining to take an application packet home. The historical probability P(X) that the interested prospective members would join is below.

0 join: 0.1

1 join: 0.2

2 join: 0.4

3 join: 0.2

4 join: 0.1

When they hold an open house this year, how many members (what is the expected value) should they anticipate joining?

Explain your approach to determining the number of expected new members in a three-page response. Be sure to research sources to support your ideas, and integrate your sources using APA-formatted citations and matching reference lists.

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In compliance with BrainMass rules this is not a hand in ready essay but is only guidance

The question to us is "what is the expected value?". The expected value is the mean of a discrete random variable. This follows from the law of large numbers. According to this law, if there is a large number of repeated trials, the average of the results will approximately equal to the expected value. The expected value is the mean value in the long run for many repeated samples. The symbol is E(X). If we follow the law, we are required to take each observed X value and multiply it by its respective probability. These products are added to reach our expected value. In some literature this is referred to as weighted average (1). The rationale is that it takes into account the probability of each outcome and weighs it accordingly. The other option is an un-weighted average which would not take into account the probability of each outcome and weigh each outcome equally.

The type of expected value we require is for a binomial random variable. It is called binomial because there are only two outcomes, the number of prospective members join or they do not. The basic expected value formula is the probability of an event multiplied by the amount of times the event happens.

The theory is that in the long run average value of repetition of an experiment it represents is the expected value. In literature the expected value is also called the expectation, mathematical expectation, EV, average mean value, mean, or the first moment(2). For the purpose of making calculations the expected value of a ...

#### Solution Summary

The response provides you a structured explanation of expected value. It also gives you the relevant references.