Dawn of the planet of Apes

Probability Level 3

Its dawn of the planet of apes and somehow an ape got into an armed spaceship. He is trying to launch a missile by clicking random buttons on the board.

There are three buttons on the board: $(R)Red$ , $(G)Green$ and $(B)Blue$ . A missile will be launched if the ape presses Red, Green, Blue consecutively in sequence. What is the expected number of missiles that he will launch if in total he pressed $1000$ buttons while in the spaceship?

Choose the closest integer as your answer.

Example:

Suppose he pressed the following buttons in sequence: $R, G, B, B, G, R, G, B, B$ , then $2$ missiles will be launched.

Assumptions:

Assume the ape is not capable of learning.
He presses the buttons equally likely
He doesn't break the buttons while pressing.

30 10 40 20

1 solution

Lokesh Sharma
Oct 11, 2016

We define two random variables:

$M$ : Let $M$ be the number of missiles launched in 1000 button pressed.

$M_i$ : Let $M_i$ be the event that ith press was $R$ , (i+1)th press was $G$ and (i+2)th press was $B$

We can say that:

$M = M_1 + M_2 + ... + M_{998}$

Using linearity of expectation :

$E[M] = E[M_1] + E[M_2] + ... + E[M_{998}]$

$E[M] = 998 * (\frac{1}{3})^{3}$

$E[M] = 36.96$ which is closest to 40

My question reg. your solution is that I can't see how necessary constraints for the random variables $M_{i}$ are considered. For example it is not possible that synchronously $M_{1}=M_{2}=M_{3}=1$ is valid.

Andreas Wendler - 4 years, 8 months ago

You are right. These constrained are not considered at all. This is the power of linearity of expectation. Even though $M_1$ is dependent on $M_2$ , we don't care.

Lokesh Sharma - 4 years, 8 months ago

I took the the expectation by using P (RGB) = (1/3)^3 = 1/27. 1000*1/27 = 37 1/27 so 40 is the nearest answer

Peter van der Linden - 4 years, 8 months ago

I calculated expected number of pressed buttons to launch a missile: E=39. So, 1000/39 = 25.6...$ , then the answer is 30.

Bogdan Kejžar - 4 years, 8 months ago

Dawn of the planet of Apes

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