Question

A bowl contains five chips, which cannot be distinguished by a sense of touch alone. Three of the chips are marked $$\$ 1$$ each and the remaining two are marked $$\$ 4$$ each. A player is blindfolded and draws, at random and without replacement, two chips from the bowl. The player is paid an amount equal to the sum of the values of the two chips that he draws and the game is over. Suppose it costs \(p_{0}\) dollars to play the game. Let the random variable \(G\) be the gain to a player of the game. Determine the distribution of \(G\) and the \(E(G)\). Determine \(p_{0}\) so that the game is fair. The \(\mathrm{R}\) code sample \((\mathrm{c}(1,1,1,4,4), 2)\) computes a sample for this game. Expand this into an \(\mathrm{R}\) function that simulates the game.

Short answer

The distribution of the gain (G) in this game is G = {2 with probability 0.6, 5 with probability 0.3, 8 with probability 0.1}. The expected gain (E(G)) is 3.1 dollars. Therefore, for the game to be fair, it should cost 3.1 dollars to play. The provided R function simulates this game.

Step-by-step solution

Determine the distribution of the Gain (G)

Since the player is blindfolded and draws two chips out of five from the bowl, there are a total of \({{5}\choose{2}}=10\) possible outcomes. There are 6 possibilities whereby the sum of the two chips equals to 2 dollars (when both drawn chips are 1 dollar each), three possibilities where the sum equals 5 dollars (one of each chip is drawn), and one possibility where the sum equals 8 dollars (both drawn chips are 4 dollars each). Therefore, the distribution of the random variable G is G ~ f(2) = 0.6, f(5) = 0.3, f(8) = 0.1

Calculate the Expected Gain (E(G))

Using the definition of Expected value for a discreet random variable, which is the sum of each outcome times its occurrence probability, we have \( E(G) = [2*0.6] + [5*0.3] + [8*0.1] = 3.1 dollars \)

Determine the Fair Game Charge

For a game to be fair, the cost of the play should be equal to the expected gain. Therefore, \( p_{0} = E(G) = 3.1 dollars \). So, the game is fair if it costs 3.1 dollars to play.

Create an R function to Simulate the Game

Now let's create a function in R that implements this game. We need a function that randomly selects two chips from the bowl and returns the sum of their values. Here is one such function:\n\n game_simulate <- function() {\n chips <- c(1, 1, 1, 4, 4) \n drawn_chips <- sample(chips, 2, replace = FALSE)\n return(sum(drawn_chips))\n }