Probability and Variance in game

[Eirny]

Let Y be a random variable indicating the number of successes out n trials with a probability of a success p.

y | n, p ~ Binomial (n, p).
Expected value = np
Variance = np(1-p)
Standard Deviation = sqrt{np(1-p)}

Example:
We use Hasty Touch (HT) 100 times with the probability of a success 0.5
Expected number of HT successes out of 100 trials = np = 100 x 0.5 = 50
Variance = np(1-p) = 100 x 0.5 (1 - 0.5) = 25
Standard Deviation = sqrt(25) = 5

Plot http://www.wolframalpha.com/input/?i...%3D100+p%3D0.5


If you do not know the probability of a success and want to quantify your uncertainty, then

y | n, p ~ Binomial (n, p)
p ~ Beta(1, 1)
p | n, y ~ Beta(1 + number of successes, 1 + number of trials - number of successes)

Examples:

20 HT, 13 successes.
Beta (1 + 13, 1 + 20 - 13) = Beta (14, 8)
Plot http://www.wolframalpha.com/input/?i...14%2C+beta%3D8

100 HT, 44 successes.
Beta (1 + 44, 1 + 100 - 44) = Beta (45, 57)
Plot http://www.wolframalpha.com/input/?i...5%2C+beta%3D57

Note how much the variances decreases by increasing the sample size.


Further Reading:
http://en.wikipedia.org/wiki/Binomial_distribution
http://en.wikipedia.org/wiki/Beta_distribution
http://en.wikipedia.org/wiki/Checkin...a_coin_is_fair
http://en.wikipedia.org/wiki/Memory_errors

[Marxam]

Let Y be a random variable indicating the number of successes out n trials with a probability of a success p.

y | n, p ~ Binomial (n, p).
Expected val...