Question

a lottery costs $1 per ticket. the player selects a sincle letter from A to T and a single digit from 0 to 9. if both the letter and the digit match the letter and digit picked on that day, the player wins $250. what is the expected value of a lottery ticket/

Answer 1

Answer:  0.25 dollars (ie 25 cents)

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Explanation:

The letter T is the 20th letter in the English alphabet. This means that there are 20 letters to pick from to go with the 10 single digit numbers to pick from. We have 20*10 = 200 different combinations (eg: one combination is K4).

Out of these 200 combinations, there is exactly one winning match.

The probability of winning is therefore P(W) = 1/200 = 0.005 and the probability of losing is P(L) = 1-P(W) = 1-0.005 = 0.995 where W and L represent winning and losing respectively.

The net value of winning is V(W) = 250 - 1 = 249 because the player wins $250 but they lose $1 (which is the cost of the ticket). The net value of losing is V(L) = -1, indicating the player has lost that $1 and hasn't gained any money at all.

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To summarize so far:

P(W) = 0.005

P(L) = 0.995

V(W) = 249

V(L) = -1

Multiply the probability P values with the corresponding net values V, then add up the products to get the expected value:

E(X) = P(W)*V(W) + P(L)*V(L)

E(X) = 0.005*249 + 0.995*(-1)

E(X) = 0.25

The expected value is 0.25 dollars or 25 cents

This positive expected value means that the game is favored toward the player, because over the long run, the player gains an average of 25 cents each time they play. On the other side of the spectrum, the lottery company loses on average 25 cents each time. The nonzero expected value means that the game is not fair.