Final answer:
The correct probability function for the total number of flips required until the first head appears in a coin flip is P(X) = (1/2)^x. For a biased coin game, the expected outcome is a loss over many games. When flipping two fair coins, probabilities of different events are calculated individually. Correct option is A) P(X) = (1/2)x, x = 1, 2, 3...
Step-by-step explanation:
The probability function for the total number of flips required until the first head appears in a coin flip is given by the formula P(X) = (1/2)x, where x is the number of flips. This represents a geometric distribution with the probability of success (getting a head) on each trial being 1/2. Therefore, the correct answer to the student's question is option A) P(X) = (1/2)x, x = 1, 2, 3...
Returns from a biased coin game can be examined by calculating the expected value. With P(heads) = 3/4 and P(tails) = 1/4, the expected payout E(X) is E(X) = (3/4)(-$6) + (1/4)($10) = -$4.50 + $2.50 = -$2. When playing many times, the expected result is a loss; hence, a player would not come out ahead.
In flipping two fair coins, different probability events include:
- Finding the probability of getting at most one tail which would be the cases of zero or one tail (F).
- Getting two faces that are the same (G).
- Getting a head on the first flip following any outcome on the second flip (H).
When considering 15 flips of a fair coin and calculating the probability of getting more than 10 heads, one would use the binomial probability formula. Since each flip is independent and the coin is fair, the probability of getting a head is 0.5.