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I have been stuck with this question for some time and I'm not sure if I'm doing it correctly.

There are three players who each have their own coin—each coin having a different probability for it to land on heads. They play this game simultaneously, the first one to flip heads wins the game.

Let $X$ be the number of flips player 1 needs to get heads, $Y$ the number of flips for player 2 till he has heads, and $Z$ be the flips for player 3 till he has heads.

Now I noted this down as $X\sim \textrm{Geom}(p), Y\sim \textrm{Geom}(q)$, and $Z\sim \textrm{Geom}(r)$. These three variables are independent.

My first question would be: What is the probability that player 1 wins? I thought it was enough to calculate $\mathrm{P}(X < Y < Z) + \mathrm{P}(X < Z < Y)$.

But now I'm not sure that this is enough.

I'm also not so sure about this: the number of turns needed to determine the winner of this game would be the $\mathrm{Min}\{X, Y, Z\}$, since the first person to flip heads will win. Hence meaning that its the minimum right?

So the number of turns after the winner is known until the last person to flip heads would be the $M = \mathrm{Max}\{X,Y,Z\}$, after someone wins the game you go and play until everyone has heads. This would mean the maximum of the flips? So since these three variables are geometric and independent this comes down to: $$ \mathrm{P} (M>m) = \mathrm{P} (X>m; Y>m; Z>m)= ((1-p)(1-q)(1-r)^m) $$

This means that $M$ is also Geometric but then moved to zero.

So I hope someone can maybe tell me if I'm doing this correctly, or if I should calculate these things differently.

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    $\begingroup$ You need to specify in a little more detail what "They play this game simultaneously, the first one to flip heads wins the game" means - what if e.g. X and Y both get a heads on their first toss? $\endgroup$ – Silverfish Jan 29 '15 at 3:15
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    $\begingroup$ Since $X, Y$, and $Z$ are claimed to be geometric random variables with different parameters, they are not iid random variables if you are using the standard meaning of iid independent and identically distributed. Clearly they are not identically distributed. Also, as silverfish asks, you need to specify what it means to be the first to toss a head in case of ties. $\endgroup$ – Dilip Sarwate Jan 29 '15 at 3:57
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    $\begingroup$ Simply ignore ties. $\endgroup$ – David G. Stork Jan 29 '15 at 5:37
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    $\begingroup$ @DavidG.Stork please allow the original poster to specify the intent of their question. $\endgroup$ – Glen_b -Reinstate Monica Jan 29 '15 at 11:37
  • $\begingroup$ Dilip: Yeah sorry about the iid, its clearly doesnt have the same parameter, its just independent. @Silverfish: Now to calculate that player 1 wins this game: You cant have two players that toss heads at the same time. So to calculate the player 1 wins: if we think about this in turns we can say that player 1 wins in the kth turn. You have to assume that he can only win if no one else flipped heads before this kth turn and that in this turn no one else has heads. This is how I understand this, I hope it makes more sense now. $\endgroup$ – Rabea Jan 29 '15 at 11:53
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I am sure there are many ways to analyze this situation, but the following one is appealing for its simplicity and use of only the most basic properties of probability. Assuming ties for heads cause the game to continue until a result is uniquely determined, it shows the chance of a player winning the game is her relative odds: the proportion of her odds of heads, divided by the sum of the odds of heads among all players. The game length obviously has a geometric distribution. Its parameter is proportional to this sum of odds. The constant of proportionality is the product of all the chances of tails.

Other methods of resolving ties (to create a definite winner) can be analyzed in a similar way, starting with computing the chance that a given player will win a particular round and continuing as shown here.


Let there be $m$ players, each using a coin with probability $p_i$ of heads. Let $i,j,\ldots, k$ be any permutation of $(1,2,\ldots,m)$.

Suppose that when a tie occurs, no outcome is declared and play continues. Then the chance that player $i$ wins is the chance that she is the sole person to toss heads, equal to

$$p_i(1-p_j)\ldots(1-p_k) = \frac{p_i}{1-p_i}\prod_{l=1}^m (1-p_l) = \pi_i Q$$

where I have written $\pi_i = p_i/(1-p_i)$ for player $i$'s odds of heads and $Q$ for the product of all $1-p_l$ (the chance that everybody simultaneously observes a tail).

If no player wins a round, the game starts over, with exactly the same probabilities. Therefore the chance that player $i$ wins the entire game is the chance they can win a given round, divided by the sum of all players' chances to win the round:

$$\Pr(i\text{ wins}) = \frac{\pi_i Q}{\sum_{l=1}^m \pi_l Q} = \frac{\pi_i}{\sum_{l=1}^m \pi_l} = \frac{\pi_i}{\pi}.$$

It is their proportion of the total odds $\pi$.

The chance that the game ends on any particular round is the chance that exactly one player observes heads, equal to

$$\sum_{l=1}^m \pi_l Q = \pi Q.$$

Thus the length of the game has a geometric distribution with parameter $\pi Q$.

This result is supported by simulation:

Figure

It was carried out with this R script, which can simulate and summarize tens of millions of tosses per second for arbitrarily many players (up to limits determined by RAM).

coins <- log(2:10); coins <- coins / sum(coins)
n.players <- length(coins)
n.rounds <- 1e6
#
# Simulate many rounds.
#
system.time({
  tosses <- matrix(runif(n.rounds * n.players), nrow=n.players) < coins
  wins <- colSums(tosses) == 1
  winners <- colSums(1:n.players * tosses) * wins
  (results <- tabulate(winners[winners != 0], n.players))
})
#
# Compare to theory.
#
odds <- coins / (1-coins)
p <- odds / sum(odds)
rbind(Simulation=results / sum(results), Theory=p)
#
# Display the distribution of game lengths compared to the geometric distribution.
#
lengths <- diff(c(0, which(wins==TRUE)))
Q <- prod(1 - coins)
pQ <- sum(odds) * Q
n.max <- ceiling((log(.002) + log(pQ)) / log(1-pQ)) # Plotting limit
prob.geometric <- (1 - pQ)^(1:n.max - 1) * pQ

subtitle <- paste(sum(results),"games played with coins", 
                  paste(round(coins, 2), collapse=","))
hist(lengths[lengths < n.max], freq=FALSE, breaks=(1:n.max)-1/2,
     main="Simulation (bars) vs. Theory (lines)",
     xlab="Game lengths",
     sub=subtitle)

invisible(sapply(1:n.max, function(i) {
  lines(c(i,i), c(0, prob.geometric[i]), lwd=2, col="Red")
}))
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