Difficulty understanding "Recursive Method" in Probablity Puzzles App

Question

You are repeatedly tossing a fair die. How many tosses until you get a 5 or a 6.

I got the answer $3$ using $P_iX_i$ that is creating a infinite series but the hint to the problem says

One approach is to sum an infinite series. Another is to notice that the problem either ends or restarts after first toss: this lets to express answer recursively, meaning you can set up an equation where answer appears on both sides

I'm not sure how to interpret the second part. It makes sense that the problem end or restarts but I don't understand what to make of it or how to solve it using recursion

I think they are just referring to the memoryless property. – Karn Watcharasupat Dec 05 '17 at 14:35 — Karn Watcharasupat, Dec 05 '17 at 14:35

score 2 · Accepted Answer · answered Dec 05 '17 at 14:38

2

Suppose the expected number of rolls is $E$. Then when a roll is made:

there is a $\frac13$ probability that a 5 or 6 is rolled, stopping the process
there is a $\frac23$ probability that a 5 or 6 is not rolled, leaving an expected $E$ more rolls to go

This gives the equation $$E=\frac13\cdot1+\frac23\cdot(1+E)$$ $$\frac13E=1$$ $$E=3$$

answered Dec 05 '17 at 14:38

Parcly Taxel

103,344

What does $(1+E)$ part mean? Because in the infinite series I made, the second term was $22/31/3$ – Anvit Dec 05 '17 at 14:45
@AnvitGarg $1+E$: one roll has been made, and an expected $E$ more rolls are left since a 5 or 6 was not rolled. – Parcly Taxel Dec 05 '17 at 14:46
What is this method called and where can I get some practice questions? – Anvit Dec 05 '17 at 17:17
@AnvitGarg I don't call this a "method", I call this a "hack", which means that its use is situational. For an extravagant example, see this answer. – Parcly Taxel Dec 05 '17 at 17:20
everything after the first 5 postulates is a hack then xD – Anvit Dec 05 '17 at 17:40

Difficulty understanding "Recursive Method" in Probablity Puzzles App

1 Answers1