What is the probability of getting
tails at least once if a coin is flipped three times?
There are lots of ways of
approaching this problem. Whichever method we decide to use,
we need to recall that each flip or toss of a coin is an independent event. The outcome of the first flip does
not affect the outcome of any others. One way of approaching this problem
would be to list all the possible combinations when flipping a coin three times. It is possible that all three coins
could land on tails. Another possibility would be to
land on tails for our first two tosses and heads on the third one. We could get two tails and a head
in two other ways. Tails, head, tails or heads, tails,
tails. Getting one tail and two heads
could happen tails, heads, heads. It could also happen heads, tails,
heads or heads, heads, tails.
Finally, all three coins could land
on heads. This means that there are eight
different combinations that could occur. We want the probability of getting
tails at least once. Our top combination has three
tails. The next three have two tails. The three combinations after this
have one coin landing on tails. This means that seven out of the
eight combinations end up with getting tails at least once. The probability of this occurring
is therefore seven out of eight or seven-eighths.
An alternative method would be to
calculate the probability of the only combination we don’t want first, the
probability of three heads. The probability of landing on heads
in any individual toss of a coin is one-half. As each of the events or tosses are
independent, we can multiply these fractions to calculate the probability of getting
three heads. The probability of three heads is
one-eighth. As the probability of getting tails
at least once is everything else, we can subtract this answer from one as we know
probabilities sum to one. Subtracting one-eighth from one,
once again, gives us an answer of seven-eighths. When flipping a coin three times,
the probability of getting tails at least once is seven-eighths.