A die that decides how many dice
You roll a fair die and read the number . Then you roll more fair dice. Find the expected total of all the dice rolled, the first one included.
Fractions, decimals with at least 3 significant digits, and expressions such as sqrt(2)/2, e^-1 or pi/4.
Condition on : given , the extra dice add on average.
Then average over , and do not forget the first die.
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s Shown after a correct answer. Revealing it first marks the problem as seen, not solved.
Idea. Condition on the first roll. Given , the extra dice contribute on average; then take the average over .
Step 1. Given , the total is plus the sum of dice, so its expected value is .
Step 2. Averaging over , with :
Remark. Without the first die, the answer is . This is the simplest case of Wald's identity: the expected sum of a random number of independent terms, with independent of the terms, is times the mean of one term.