Expectations

Probability Level 4

You roll a standard six-faced-die and get $n.$ Then, you roll it again until you get a number greater than or equal to $n$ .

The expected number of times you need to roll your die (after the initial roll) is of the form $\dfrac{A}{B}$ where $A$ and $B$ are co-prime positive integers.

Find $A+B.$

The answer is 69.

1 solution

Snehal Shekatkar
Oct 26, 2014

Note that for any value $n$ that we got on first trial, the probability that we get number greater than or equal to $n$ is $\frac{7-n}{6}$ . Thus on an average, we need $\frac{6}{7-n}$ attempts to get number greater than or equal to $n$ . Hence the average number of attempts when all $n$ are considered is:

$\frac{1}{6}\sum_{n=1}^{6}\frac{6}{7-n} = \frac{1764}{720} = \frac{49}{20}$

Hence, $\boxed{A+B = 49+20 = 69}$

Expectations

The answer is 69.

1 solution

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