Final answer:
The probability of a fair coin landing heads up 11 times in a row is (0.5)^11. It is not the same as the probability of it not landing tails up 11 times, which is 1 - (0.5)^11, because the latter includes all other combinations of outcomes over eleven coin tosses.
Step-by-step explanation:
The question is whether the probability of getting 11 heads in a row when tossing a coin is the same as the probability of not getting 11 tails in a row. When tossing a coin, there are only two possible outcomes: heads or tails, and if the coin is fair, each outcome has an equal chance of occurring. Therefore, the probability of getting heads is 0.5 and the probability of getting tails is also 0.5 at each toss.
For the coin to land heads up all eleven times, we have to multiply the probability of getting a head on a single toss (0.5) by itself eleven times: (0.5)^11. This is because each toss is an independent event, and the outcome of one toss does not affect the next. The result is the probability of getting heads 11 times in a row.
Now, when we consider the probability of the coin NOT landing tails up all eleven times, this actually includes all possible outcomes except the one where it lands tails up 11 times. This includes the scenario described in the question, where the coin lands heads up all eleven times, among all other combinations. To find this probability, we subtract the probability of getting tails 11 times in a row from 1, which gives us 1 - (0.5)^11.
Thus, the probability of the coin landing heads up all eleven times is incredibly small but not the same as the probability of it NOT landing tails up all eleven times, which is substantially larger because it encompasses all other combinations of heads and tails across eleven tosses.