Determine a casino's expected net income from a 24-hour period at a single roulette table if the casino's total overhead for the table is $50 per hour and if customers place a total of $7,000 on single-number bets, $4,000 on two-number bets, $4,000 on four-number bets, $3,000 on six-number bets, $7,000 on low-number bets, and $8,000 on red-number bets. (Assume the expected value of each of these $1 bets in roulette is -$0.053.)
To determine the casino's expected net income from a 24-hour period at a single roulette table, we need to calculate the total amount wagered and subtract the expected value of each bet.
Step 1: Calculate the total amount wagered:
Total amount wagered = $7,000 (single-number bets) + $4,000 (two-number bets) + $4,000 (four-number bets) + $3,000 (six-number bets) + $7,000 (low-number bets) + $8,000 (red-number bets)
Total amount wagered = $33,000
Step 2: Calculate the total expected value of all the bets:
Expected value of each $1 bet = -$0.053 (this value is already given)
Expected value of all bets = (Total amount wagered) * (Expected value of each $1 bet)
Expected value of all bets = $33,000 * (-$0.053)
Expected value of all bets = -$1,749.00
Step 3: Calculate the total overhead cost for the 24-hour period:
Total overhead cost = $50/hour * 24 hours
Total overhead cost = $1,200.00
Step 4: Calculate the casino's expected net income:
Expected net income = (Total amount wagered) + (Total overhead cost) + (Expected value of all bets)
Expected net income = $33,000 + $1,200.00 + (-$1,749.00)
Expected net income = $32,451.00
Therefore, the casino's expected net income from a 24-hour period at a single roulette table is $32,451.00.