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?
We'll be exploring the world of probability and statistics through an experiment involving coin tossing. The experiment is simple, we'll be tossing a fair coin until a head appears twice in a row. But what is the sample space for this experiment and what is the probability that the coin will be tossed exactly four times before getting two heads in a row? First, let's examine the sample space for this experiment. Since the coin can either be heads or tails, the sample space for the first toss is {H, T}. For the second toss, the sample space is {HH, HT, TH, TT}. However, the experiment stops as soon as two heads appear in a row, so the sample space for the experiment is {HH, HT, TH, TTT...}, where T can be repeated any number of times. Now, let's calculate the probability that the coin will be tossed exactly four times. Since the coin is fair, the probability of getting a head is 1/2 and the probability of getting a tail is 1/2. The probability of getting four tosses and not getting two heads in a row is (1/2)^4 = 1/16. In conclusion, through this simple experiment, we've explored the concept of sample space and probability in coin tossing. The experiment gives an insight on how different outcomes can be obtained and the probability of getting those outcomes. The importance of understanding the sample space and probability can be applied in various real-world situations as well.
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