A coin is tossed 100 times , Find the probability that tail occurs odd number of times!
I do not know the answer, but I tried this, that there are these $4$ possible outcomes in which tossing of a coin $100$ times can unfold.
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head occurs odd times
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head occurs even times
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tail occurs odd times
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tail occurs even times
Getting a head is equally likely as getting a tail, similarly for odd times and even times.
Thus, all of these events must have same the probability, i.e. $\dfrac{1}{4}$.
Is this the correct answer? Is there an alternate way of solving this problem? Lets hear it!
Best Answer
Suppose you flipped heads an even number of times on the first 99 flips. Then there is a $\frac{1}{2}$ probability that you will get another heads, and thus and odd number of heads total. So in this case it's 50-50.
Suppose you flipped heads an odd number of times on the first 99 flips. Then there is a $\frac{1}{2}$ probability that you will get another heads, and thus and even number of heads total. So in this situation it's also 50-50.
So regardless of what happens for the first 99 flips, there's a $\frac{1}{2}$ change you end up with an odd number of heads and a $\frac{1}{2}$ chance you end up with an even number of heads.