Final answer:
In the probability space (omega, F, P) for tossing a biased coin three times, the sample space omega includes eight possible outcomes that factor in the bias, the event space F comprises subsets of omega, and the probability measure P assigns probability values to these events. The expected value calculation using the example probabilities indicates a player would not come out ahead playing this biased coin game.
Step-by-step explanation:
When considering the probability space (ω, F, P) for the experiment of tossing a biased coin three times, we first need to identify the sample space ω, which consists of all possible outcomes. With a biased coin, the probability of heads (H) and tails (T) are not equal. For this case, let's say the probabilities are P(H) = 3/4 and P(T) = 1/4, as an example. The sample space ω for three tosses would comprise the following eight elements: {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
The event space F is the set of all events, where an event is a subset of the sample space ω. For instance, an event could be {HHH, HHT}, representing the outcome of getting at least two heads in three tosses.
The probability measure P assigns probabilities to the events in F. For instance, since the coin is biased, calculating the probability of each element of the sample space would require raising the probability of heads or tails to the respective power corresponding to the number of times it appears in a particular outcome. For example, P(HHH) would be (3/4)^3, P(HHT) would be 3/4 * 3/4 * 1/4, and so on.
We can also consider whether the expected value from the game of tossing this biased coin indicates if you would come out ahead. We calculate the expected value by multiplying the payout (or loss) by the probability for each outcome and sum these values across all outcomes. With P(H) = 3/4, if you toss a head, you pay $6, and if you toss a tail, you win $10. The expected value per game is calculated as E(X) = P(H)*(-$6) + P(T)*($10) = -4.5 + 2.5 = -$2. Therefore, if you play this game many times, you will not come out ahead in the long run.