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As part of a bigger game-theory problem, I've been trying to solve a rather simple probability question, and I seem to be getting the wrong answers. Here's the problem:

Dice are rolled to determine a pass or fail, and the probability of either is equal. If the roll is a pass, the entire condition passes, and if it is a failure it is re-rolled until either it passes, or has failed 3 times. What is the probability of the entire condition passing?

The way I solved this is by listing all possible states, of which there are four:

fail fail fail = failure
fail fail pass = success
fail pass      = success
pass           = success

based on this, the probability of overall success is $\frac{3}{4}$, or 0.75.

To test it, I wrote a simple simulation in python as follows:

def test():
    fails = 0
    while fails < 3:
        if random.choice([True, False]):
            return True
        else:
            fails += 1
    return False


passes = 0
for i in range(1000000):
    if test():
        passes += 1

print(passes / 1000000)

It shows that the probability is 0.875. I'm sure that the simulation is corrent - I rewrote it in several different ways and came up with the same answer each time. So where am I going wrong with the mathematics?

The followup is to generalise this slightly where the probability of a pass or fail roll is not equal (e.g. only a 6 of a 6-sided die means pass). I have no idea how to solve that though.

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    $\begingroup$ You've made the classic probability mistake of using a sample space where not all of the elements are equally likely: (fail, pass) is actually twice as likely as (fail, fail, pass) or (fail, fail, fail). $\endgroup$ May 5, 2021 at 6:45
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    $\begingroup$ "based on this, the probability of overall success is $\frac{3}{4}$, or 0.75." only if each of the four outcomes is equally likely. $\endgroup$
    – Arthur
    May 5, 2021 at 6:45

2 Answers 2

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To see why the probability is $\frac78,$ let's slightly rephrase the question: let's say instead of simulating the flips until we get a success (let's say heads) let's just flip the coin three times straight away and then look at the results. A set of three flips is successful if and only if there is at least one heads in the three flips, so the probability of a success must be equal to the probability of getting at least one heads.

So, letting $X \sim B(3, 0.5)$ be the number of heads, we want $\text{P}(X \geq 1).$ By the law of complements, this is equal to $1 - \text{P}(X = 0) = 1 - \left(\frac12\right)^3 = \frac78.$

Alternatively, using a sample space, there are $2^3 = 8$ possible sequences of three flips, and only $1$ has all three tails, so the other $7$ have at least one heads for a probability of $\frac78.$

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Say $1, 2, 3$ are considered success and $4, 5, 6$ are considered failure. So we have equal probability of success or failure in each roll of the dice. As we have up to $3$ rolls, the probability of success is,

$P(S) = 0.5 + 0.5 \times 0.5 + 0.5^2 \times 0.5 = 0.875$

First term is the probability of success in one roll, second term is the probability of success in two rolls (Failure, Success) and last term is the probability of success in three rolls (Failure, Failure, Success).

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