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You flip a fair coin 576 times. Without using a calculator, calculate the probability of flipping at least 312 heads.

What we want is \(P(X > 311) = 1 - P(X<=311)\). Flipping coins follows a Binomial. Through the Central Limit Theorem, we can approximate the Binomial with a normal distribution \(N(np, np(1-p)\). Specifically for our case:

\[ X\sim N\big(np, np(1-p)\big) = N(288, 144) \]

The standard deviation of the normal approximation is \(\sigma=12\). Hence, we can get:

\[ 312 - 288 = 24 = 2 \sigma \]

We know that the probability of getting an observation of at least 2 standard deviations from the mean of a normal distribution is roughly 0.025, and hence \(P(X > 311) = 0.025\).

For more info regarding the normal approximation see: https://en.wikipedia.org/wiki/Binomial_distribution#Normal_approximation

Taken from the above link: “

This approximation, known as de Moivre–Laplace theorem, is a huge time-saver when undertaking calculations by hand (exact calculations with large n are very onerous); historically, it was the first use of the normal distribution, introduced in Abraham de Moivre’s book The Doctrine of Chances in 1738. Nowadays, it can be seen as a consequence of the central limit theorem since B(n, p) is a sum of n independent, identically distributed Bernoulli variables with parameter p. This fact is the basis of a hypothesis test, a proportion z-test, for the value of p using x/n, the sample proportion and estimator of p, in a common test statistic.”

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