If nine fair coins are tossed, using the complement formula, what is the probability of obtaining at least one head and at least one tail?
Two fair coins are tossed and at least one is tail. What is the probability that both coins are tails?
To find the probability of obtaining at least one head and at least one tail when nine fair coins are tossed, we can use the complement rule.
The complement rule states that the probability of an event occurring is equal to one minus the probability of the event not occurring.
In this case, the complement of "getting at least one head and at least one tail" is "getting all heads or all tails."
To calculate the probability of getting all heads, we need to consider that there are two possible outcomes for each coin toss (head or tail), and we want all nine coins to land on heads. So the probability of getting all heads is (1/2)^9, since for each coin toss the probability of getting heads is 1/2.
Similarly, the probability of getting all tails is also (1/2)^9.
Now, the probability of either getting all heads or all tails (the complement event) is the sum of the probabilities of these two exclusive events, which is (1/2)^9 + (1/2)^9.
Therefore, the probability of obtaining at least one head and at least one tail is 1 - [(1/2)^9 + (1/2)^9].
Calculating the value, we have:
(1/2)^9 ≈ 0.001953125
So, 1 - [(1/2)^9 + (1/2)^9] = 1 - [0.001953125 + 0.001953125] = 1 - 0.00390625 = 0.99609375.
Therefore, the probability of obtaining at least one head and at least one tail when nine fair coins are tossed using the complement formula is approximately 0.9961 or 99.61%.