Expected Value and Binomial Distribution

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What is Expected Value?

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The Expected Value (EV) is a measure of the center of a probability distribution, calculated as the sum of all possible values, each multiplied by its probability of occurrence.

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How do you calculate Expected Value?

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EV = Σ (X * P(X)), where X is the value of the random variable and P(X) is the probability of X.

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What is a Binomial Distribution?

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A Binomial Distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states across a number of trials.

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What are the parameters of a Binomial Distribution?

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The parameters are n (number of trials) and p (probability of success on each trial).

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What is the formula for the probability of getting exactly k successes in a Binomial Distribution?

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P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))

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What does 'n choose k' mean?

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'n choose k' or C(n, k) is the number of ways to choose k successes from n trials, calculated as n! / (k!(n-k)!).

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What is the probability of at least k successes in a Binomial Distribution?

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P(X ≥ k) = 1 - P(X < k) = 1 - Σ P(X = i) for i = 0 to k-1.

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What is the significance of the Central Limit Theorem in relation to Binomial Distribution?

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The Central Limit Theorem states that as n increases, the distribution of the sample mean will approach a normal distribution, regardless of the shape of the population distribution.

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How do you interpret the result of an Expected Value calculation?

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The Expected Value provides a long-term average outcome of a random variable, helping to inform decisions based on probabilities.

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What is the relationship between Expected Value and gambling?

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In gambling, the Expected Value can indicate whether a game is favorable or unfavorable based on the probabilities and payouts.