Final answer:
To calculate the expected payoff of the dice game, one must consider all possible dice sum outcomes and their probabilities, multiply by the payout rate ($7 times the sum), and subtract the cost ($49). However, since the maximum payout ($84) is lower than the cost, it is evident that the expected payoff will be less than $49, indicating the game is not fair.
Step-by-step explanation:
The game in question involves rolling a pair of dice and then multiplying the sum of the numbers rolled by $7. To determine the expected payoff of the game, one has to calculate the average amount one would expect to win or lose per roll over a long series of games.
First, we ascertain all the possible sums of the dice, which can range from 2 (1+1) to 12 (6+6). Then we find the probability of each sum occurring and multiply it by $7 (since the payout is 7 times the sum), and subtract the cost of the game, which is $49.
The sum of 2 has a 1/36 chance of being rolled, sum of 3 has a 2/36 chance, and this pattern continues up to the sum of 7, which has the highest chance of 6/36, after which the pattern reverses down to a 1/36 chance of rolling a 12. By multiplying each sum by its respective probability, multiplying this by $7, adding these values together, and subtracting the $49 cost, we derive the expected value of the game.
However, without calculating all individual expected values for each sum, we can deduce that because the maximum sum (12) would yield a maximum payout of $84 ($7*12), which is still less than the cost ($49), the game cannot be fair. A fair game would require that the expected payoff be equal to the cost of playing the game.