Question

Show the observed probabilities for 10 trials and 100 trials for a game of Rock / Paper / Scissors with the probability of each outcome being equal.

Answer 1

Here we have given the game for Rock, Paper, and Scissor.

This game has the sample space means a collection of all the possible outcomes of an experiment is given by

S = { Rock , Paper , Scissor }

and all these outcomes are equally likely means they have an equal chance of getting any of one them.

Therefore

P ( {Rock } ) = 1 / 3 , P ({Paper}) = 1 / 3 and P ({Scissor }) = 1 / 3

Now we perform the 10 trials of this experiment then the distribution of this experiment is given by

Let X be a random variable denote an event of getting one of the outcomes of the game.

= 0 , otherwise

this is the probability distribution of getting one of the outcomes of the game.

Here we have to show that the probability of getting any particular outcome at any given draw is equal to the probability of getting each outcome.

Now let us consider an example of getting the rock on the 8th trial is = 1 / 3

and the probability of getting each outcome is P ( {Rock } ) = P ({Paper}) = P ({Scissor }) = 1 / 3

Both the above probabilities are always equal for any draw.

Now we consider for the 100 trials similarly the probability distribution in this case is

= 0 , otherwise

Here also we have to show the same condition.

Therefore let us consider the 55th  draw.

Hence Probability of getting the Paper on the 55th trial = 1 / 3

also, the probability of getting each of them is equal

that is  P ( {Rock } ) = P ({Paper}) = P ({Scissor }) = 1 / 3

Hence this is true for any trial out of given 100 trials.

Hence here we show that

The probability of getting any particular outcome at any given trial is equal to the probability of getting each outcome.