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Expected Value

If you play roulette, are you going to break-even? Maybe sometimes, but not if you play forever, because the expected value is negative. The expected value finds the average outcome of random events.

Expected Value: Level 5 Challenges

         

There are 3 kegs of water. The first keg has 1 liter, the second keg has 2 liters, and the third keg has 3 liters of water in it.

Each day our traveler picks one keg at random with each keg having the same chance of being picked, and drinks one liter of water from it. When he empties a keg, he throws it away. When he is left with one keg, he records the amount of water in it.

What is the expected amount of water (in liters) remaining in that last keg?

If this expected value can be expressed as \( \dfrac ab\), where \(a\) and \(b\) are coprime positive integers, enter \(a+b\).

\(\)
Note: You may use a calculator if you want.

Bonus: Solve this problem for 5 kegs.


Inspired by bobbym.

How long does it take you to roll a Yahtzee?

The dice game Yahtzee is a classic. Each player takes turns rolling 5 dice, 3 times each. After each roll, the player can set aside some dice that they want to "keep" and reroll the rest in order to get a better score. The best roll that someone can have is called a "Yahtzee," or 5 dice of the same number, e.g. five 1s, five 2s, etc.

It's easy to determine the probabilities of getting a Yahtzee in 1 roll. You have 5 dice, and a dice has 6 sides, so the chance of getting all 5 the same is \(\left(\frac{1}{6}\right)^5\), but since there are 6 ways to get a Yahtzee, the probability is really \(\left(\frac{1}{6}\right)^4 = \frac{1}{1296}\). From this, we can determine that one should expect to roll 5 dice 1296 times before getting a Yahtzee, right?

If you've ever played, you'll realize that the chances of getting a Yahtzee in a game are higher than that. This is because of the rule that you can keep some dice aside after each roll. If after each roll you decide to keep the dice aside that have the number that shows up the most, given an infinite number of rolls (not limited to 3 rolls as in real games), what is the expected number of turns it should take you to get a Yahtzee?

Round your answer to the nearest whole number.

Example sequence of rolls:

  • Roll 1: 1, 2, 4, 4, 5. Keep the two 4s.

  • Roll 2: 4, 4, 6, 3, 3. Keep the two 4s.

  • Roll 3: 4, 4, 1, 1, 1. Keep the three 1s.

  • Roll 4: 1, 1, 1, 5, 1. Keep the four 1s.

  • Roll 5: 1, 1, 1, 1, 1. Done after 5 rolls.

I toss a fair coin 100 times in a row. If the expected number of times that two Heads are tossed consecutively is equal to \(N\), what is the value of \(100N\)?

Clarification: A run of three or more Heads in a row will contribute more than one instance of consecutive Heads. If three Heads are tossed in a row, then two Heads have been tossed consecutively twice. If six Heads are tossed in a row, then two Heads have been tossed consecutively five times, and so on. For example, two Heads have been tossed consecutively 11 times in the following sequence of 30 tosses:

\[ THHTHHHHTHHHTHTHHHHHTTTTHHTHTT \]

A square with a unit length has two arbitrary points chosen inside it. What is the average distance between the two points?

If this can be represented in the form \(\dfrac {a + \sqrt{b} + c \ln (d+\sqrt{e})}{f}\), where \(a, b, c, d, e, f\) are positive integers, and the fraction is simplified as far as possible, find \(a+b+c+d+e+f\).

Alice and Bob are playing a coin flipping game. Each of them will flip their own quarter 100 times and note the outcome of each flip on a line, in their own scoresheet: \(H\) for heads and \(T\) for tails.

If on any line Alice marks \(HH\), she will give herself 1 point, begin a new line, and continue flipping.

If on any line Bob marks \(HT\), he will give himself 1 point, begin a new line, and continue flipping.

After 100 flips each, who is more likely to have more points?

Clarification: Here is an example of a possible scoresheet for Alice after 25 flips, where she scores 3 points.

  • \(HTT \boxed{HH}\)
  • \(HTTTTHTHTHTT \boxed{HH}\)
  • \(TTT\boxed{HH}\)
  • \(T\)

Taken from a numberphile video

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