Pete is considering placing a bet on the NCAA playoff game between Indiana and P

Question

Pete is considering placing a bet on the NCAA playoff
game between Indiana and Purdue. Without any further
information, he believes that each team has an equal chance
to win. If he wins the bet, he will win $10,000; if he loses,
he will lose $11,000. Before betting, he may pay Bobby
$1,000 for his inside prediction on the game; 60% of the
time, Bobby will predict that Indiana will win and 40% of
the time, Bobby will predict that Purdue will win. When
Bobby says that IU will win, IU has a 70% chance of
winning, and when Bobby says that Purdue will win, IU has
only a 20% chance of winning. Determine how Pete can
maximize his total expected profit. What is EVSI? What is
EVPI?

Explanation / Answer

Expected value = Probability of win * Value of win + Probability of loss * Value of loss

= 0.5 * $10,000 - 0.5 * $11,000 = -$500

As, Expected value of bet is negative, the maximum EMV is for not to bet with exepected value as 0

Expected value without Sample information = 0

Probability that Bobby will predict that Indiana will win, P(Bi) = 0.6

Probability that Bobby will predict that Purdue will win, P(Bp) = 0.4

Let Probability that IU will win is P(i) and Purdue will win be P(p)

then, P(i | Bi) = 0.7 and P(p | Bp) = 1 - 0.2 = 0.8

Expected value when Bobby will predict that Indiana will win = 0.7 * $10,000 - 0.3 * $11,000 = $3700

Expected value when Bobby will predict that Purdue will win = 0.8 * $10,000 - 0.2 * $11,000 = $5800

Expected value with Sample information = P(i) * $3700 + P(p) * $5800

= 0.6 * $3700 + 0.4 * $5800 = $4540

Expected vaue of sample information, EVSI = Expected value with Sample information - Expected value without Sample information = $4540 - 0 = $4540

As, Expected value of bet is negative, the maximum EMV is for not to bet with exepected value as 0

Expected value without Perfect information = 0

Expected value with Perfect information= Probability of win * Maximum payoff + Probability of loss * Maximum payoff (by not betting)

= 0.5 * $10,000 - 0.5 * 0 = $5,000

EVPI = Expected value with Perfect information - Expected value without Perfect information

= $5,000 - 0 = $5,000

To maximize his total expected profit, and EVSI > $1,000 (Pay for Bobby), the best alternative is to hire Bobby to know his inside prediction.

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