Question

. A coin is tossed five times. Then the probability that at least one tail occurs is​

Answer 1

Questions like this can feel daunting at times. This is especially true when you know the direct approach to solving it. Sometimes, though, it is better to look at what DOESN’T work and go from there to make your life easier.

If we want at least 1 tails, then we are allowed 1,2,3,4,
1
,
2
,
3
,
4
,
or all 5
5
as tails to show up in our tosses. The only outcome that doesn’t work for what we are looking for is getting all 5
5
tosses resulting in heads.

We know the overall probability of the tosses (all possible outcomes) must equal 1
1
. So if we consider that (on a fair coin) our chance of getting heads is (1/2)
(
1
/
2
)
per flip. The outcomes are independent so we multiply 5
5
times (1/2)5
(
1
/
2
)
5
or (1/2)(1/2)(1/2)(1/2)(1/2).
(
1
/
2
)
x
(
1
/
2
)
x
(
1
/
2
)
x
(
1
/
2
)
x
(
1
/
2
)
.


We get (1/32)
(
1
/
32
)
as out chance of getting all heads. Since we want at least 1
1
tails we subtract that from the overall probability to find our final answer.

1−(1/32)=(31/32)
1
−
(
1
/
32
)
=
(
31
/
32
)
to get at least 1 tails.

It is worth noting you can tackle this problem head on by calculating the chances of each possible way of getting tails and adding them up….but this way is much less tedious. The trick is recognizing when to work directly and when to work backwards so to speak.

Hope this helps.
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