# Find the probability that 3rd, 4th and 5th tosses are identical

A fair coin is tossed $10$ times, the tosses being indipendent of each other. I have to find the probability that 3rd, 4th and 5th tosses are identical.

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Hint: Two possibilities (i) three heads and (ii) three tails. What is the probability of $3$ consecutive heads? Of three consecutive tails? Add.
The fact that there were $10$ tosses is irrelevant. And the fact that $3$, $4$, and $5$ are next to each other s irrelevant. We would have the same answer to "what is the probability the the $2$nd, $5$th, and $10$th tosses are identical?"
probability of three consequtive head is $1/8$, and same for the tails, hence $1/4$? – La Belle Noiseuse Oct 4 '12 at 8:58
Yes, that is correct.${}{}{}{}{}$ – André Nicolas Oct 4 '12 at 9:00
Alternately, we don't care what the $3$rd toss was. The probabilty the $4$th matched it is $(1/2)$, and the probability the $5$th matched that is $(1/2)$, for a probability of $(1/2)(1/2)$. Simpler! But it is useful to get accustomed to dividing things into cases. – André Nicolas Oct 4 '12 at 9:04