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Many historians of probability note the emergence of a theory of probability as linked to humans attempts to understand games of chance. We begin with an early example called Senet that is at least as old as the year 3000 BC. Here, players moves are decided by dropping 4 paddles with two colored faces. The players are allowed to move based on the number of colored paddles that are facing up. Our goal is to explore the likely outcomes of the drops to begin to understand probability.
The table below begins to explore these possible combinations. We will look at further examples with more possible combinations but this will get us started.
|sticks||# ways to get 0 white||1 white||2 white||3 white||4 white||5 white|
The example above is one example of a more general kind of distribution of events. We have 4 two-sided sticks, and we will treat these as 4 Bernoulli Trials. To determine the probability of an outcome, we must:
As a function, we would have inputs that determine the outcome (success or failure, red or white, heads or tails, etc.), number of successes and failures, and the probability of a success. Below we have the functional form:
We unpack this as follows:
from scipy.special import comb
#ways to pick 2 things from 4 comb(4, 2)
#prob of success to the number of successes (1/2)**2
#prob of failures to number of failures (1/2)**2
#put it all together n_choose_k = comb(4, 2) p_to_the_k = (1/2)**2 one_minus_p_to_the_n_minus_k = (1/2)**2 #probability of getting 2 white sticks from four n_choose_k*p_to_the_k*one_minus_p_to_the_n_minus_k
Suppose we instead play the game of Sennet using 5 sticks.