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The classic one is the coin toss problem. How many head and how many tails on a certain number of tosses.
For example we toss a coin 4 times and call a success the result of the coin landing with the head showing. The binomial variable is the number of heads n the four coin tosses, which can take on the values 0, 1, 2, 3, or 4. The probabilities of the possible outcome can be calculated using the binomial theorem.
Here they are:
P(X = 0) = 1/16

P(X = 1) = 1/4

P(X = 2) = 3/8

P(X = 3) = 1/4

P(X = 4) = 1/16

P = 0.5n (n!/(n-x)!x! )

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