A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times?
When conducting an experiment where a coin is tossed until a head appears twice in a row, the sample space, or the set of all possible outcomes, can be represented by a sequence of H's and T's. For example, one possible outcome could be "HTTTHHH".
To determine the sample space for this experiment, we must consider the different ways in which the sequence of H's and T's can be arranged. The first head can be either a T or an H, and the second head can also be either a T or an H. Therefore, the sample space can be represented by the set of all possible sequences of H's and T's, such as {HTTTHHH, TTHHTTT, HHTTHTT, etc.}
If the coin is fair, the probability of getting a head on any individual toss is 0.5, and the probability of getting a tail is also 0.5. Therefore, the probability of getting exactly four tosses before getting two heads in a row is
(0.5 * 0.5 * 0.5 * 0.5) = 0.0625
To sum up, The sample space of the experiment is all possible sequences of H's and T's, and if the coin is fair, the probability of it being tossed exactly four times before getting two heads in a row is 0.0625.
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