Final answer:
The question is about the mathematics of probability and expected value in coin flipping scenarios. To summarize, with a biased coin and given payout structure, the expected value is negative, indicating a loss over many plays.
Step-by-step explanation:
The question deals with the mathematics subject, specifically with probability and expected value in the context of flipping coins. When flipping a fair coin three times, each outcome (head or tail) is independent of the others, and the probability of either side is 0.5 per flip. This concept is crucial in determining probable outcomes when the coin is flipped multiple times.
Considering a biased coin where the probability of heads (P(heads)) is thrice the probability of tails (P(tails)), the payouts of the game are -$6 for heads and $10 for tails. To find out if you'll come out ahead by playing the game repeatedly, you can calculate the expected value (EV). EV incorporates both the payout and the corresponding probability.
For example, if P(heads) = 3/4 and P(tails) = 1/4, the EV would be calculated as follows:
EV = (P(heads) x Payout for heads) + (P(tails) x Payout for tails)
EV = (3/4 x -$6) + (1/4 x $10)
EV = -($4.50) + $2.50
EV = -$2
As the expected value is negative, you would, on average, lose money by playing this game repeatedly.