- #1

- 120

- 6

## Homework Statement

A fair die is cast until a 6 appears. What is the probability that it must be cast more than five times.

## Homework Equations

The die is

*fair*, hence like most of the problems I can assume equally likely outcomes.

[itex]P(A^c)=1-P(A)[/itex] for any event A

## The Attempt at a Solution

Theoretically, a 6 may never come up. It should be better to calculate the complement of the event.

The complement, I think, is the event in

*at least*5 tosses, a 6 occurs. So a 6 may occur in the 1st toss or the 2nd toss or... or the 5th toss.

So I have [itex] \frac 16 + \frac 56 \cdot \frac 16 + .... + (\frac 56)^4 \cdot \frac 16 [/itex]

Then factoring out [itex]\frac 16[/itex] and writing the probabilities as a summation, I have

[itex] \frac 16 \cdot \sum_{k=1}^5 (\frac 56)^{5-i} [/itex]

Is this correct? I don't have the answer. I suspect this can be derived from a probability distribution. If my suspicions are correct, which probability distribution?