01
Summation of three: Rolling mixed dice
You have three dice. One has 4, one has 6, and one has 12 sides.
How many 4s do you expect to see if you roll these dice together?
Solution
01
(1) Call the
-sided, -sided, and -sided dice, dice respectively. Let be the event that dice rolls a . Then
, , and .
(2) Let
be the number of ‘s that are rolled. Then . Thus, by linearity of expectation.
Thus, we expect to see
Link to original‘s if we roll these dice together.
02
Jumble of coins
In my pocket I have a jumble of coins: 5 dimes, 4 quarters, 3 nickels, 3 pennies, and one big 50
-piece. I draw three at random. What is the expected value of the three?
Solution
02
(1) Let
be the value of the first coin drawn, let be the value of the second coin drawn, and let be the value of the third coin drawn. The central trick to efficiently solve this problem is to notice that are all identically distributed. One can see this by the following argument: using an ordered triple
, write down all possible permutations of drawings. Notice that the number of triples where is a dime is equal to the number of triples where is a dime is equal to the number of triples where is a dime. We can further extend this observation to all the values. Thus, the distributions of
are all the same.
(2) Another, nicer, argument is to notice that we can swap
and in these ordered triples without changing the overall set, and similarly for and there exists a bijection between ; ; and identical distribution. Thus, we have that .
(3) Let
be the sum of the values of the three coins. Then, Now,
Thus,
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03
Counting flip flops
A bag contains 50 marbles, 30 blue and 20 red. A sequence of zeros and ones is created by pulling the marbles out one at a time (without replacement) and writing a 1 if the marble drawn is blue and a zero if it is red.
How many pairs of adjacent digits in the sequence are expected to differ from each other?
Hint: Use a sum of 49 indicators.
Solution
03
(1) Let
be a sequence of indicators where is the event that the -th entry differs from the -th entry. By a similar argument to above, the are identically distributed for each , and are, in fact, independent.
(2) Let
be the number of pairs of entries that differ from each other. Then . By the above,
(3) Let
denote the -th entry. Now, Thus,
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