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Q: An urn contains lotto tickets numbered 1 to 100. You play the game by paying a fee. The rules of the game are as follows, you need to draw two tickets, you are paid the value of the lower number of the tickets. What is a fair price to play this game?

Probability Theory: The Logic of Science

A: The first ticket drawn could be any of the tickets from 1 to 100. The second draw sets up the payoff which will be the lower of the two. Assume the first drawn ticket has the number \(i\) on it. The second ticket drawn could be either greater or lesser with probabilitie as shown in the figure below.

If the chosen number is lesser, the expected value is \(\frac{i-1}{2}\). Likewise, if the chosen value is greater the expected value is \(\frac{N-i}{2}\). Thus, the total expected value can be computed as

$$

E = \frac{i-1}{N}\times \frac{i-1}{2} + \frac{N-i}{N}\times\frac{N-i}{2}

$$

Note, the above expression is just for the case when \(i^{th}\) ticket is selected. This happe…

Q: An urn contains lotto tickets numbered 1 to 100. You play the game by paying a fee. The rules of the game are as follows, you need to draw two tickets, you are paid the value of the lower number of the tickets. What is a fair price to play this game?

Probability Theory: The Logic of Science

A: The first ticket drawn could be any of the tickets from 1 to 100. The second draw sets up the payoff which will be the lower of the two. Assume the first drawn ticket has the number \(i\) on it. The second ticket drawn could be either greater or lesser with probabilitie as shown in the figure below.

If the chosen number is lesser, the expected value is \(\frac{i-1}{2}\). Likewise, if the chosen value is greater the expected value is \(\frac{N-i}{2}\). Thus, the total expected value can be computed as

$$

E = \frac{i-1}{N}\times \frac{i-1}{2} + \frac{N-i}{N}\times\frac{N-i}{2}

$$

Note, the above expression is just for the case when \(i^{th}\) ticket is selected. This happe…