Final answer:
Martin can find the probability of getting 1 head and 3 tails on 4 coin tosses by identifying the combinations and calculating the probability as 4 ways out of 16 possible outcomes, which is 0.25. For a biased coin game, calculate the expected value based on payoffs and probabilities to determine if one will come out ahead.
Step-by-step explanation:
Martin is asked to find the probability of getting 1 head and 3 tails on 4 coin tosses. To solve this, we need to understand the concept of combinations and probability. There are 4 unique positions the single head can occupy in a sequence of 4 tosses. So, for each unique position of the head, there will be 3 tails. Therefore, there are 4 ways to achieve this outcome, and since there are 24 or 16 possible outcomes in total, the probability is 4/16 or 0.25.
For a biased coin, the expected outcome of a game is calculated by multiplying the probability of each outcome by its payoff, taking into account whether you win or lose money in that outcome. Finally, you add the expected values to find the overall expected value of the game. If the expected value is positive, you will come out ahead in the long run.
With the example of the Belgian one euro-coin, P(H) is calculated by dividing the number of heads by the total number of spins, and P(T) is found by dividing the number of tails by the total number of spins. Then, you can create a probability tree to calculate the probability of getting exactly one head in two spins and the probability of getting at least one head.