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What is the expected number of consecutive Heads from 3 coin tosses?

I see the solution available here but am unsure if it is applicable to the question above, as that solution refers to a particular number of consecutive heads whereas this one could have 2 or 3 consecutive heads.

Thank You

Community
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Jojo
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    There are only 8 possible outcomes for three coin tosses. Count the number of consecutive heads in each possible outcome and average them. – MJD Oct 16 '15 at 12:52
  • @MJD So is it (2(1/8) + 2(1/8) + 3*(1/8))/3 = 7/24? – Jojo Oct 16 '15 at 13:16
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    What are you counting? How many "consecutive heads" are there in HTH? – Graham Kemp Oct 16 '15 at 13:31
  • @GrahamKemp I counted HHH, HHT and THH which have 3 consecutive Heads and 2,2 each. Am I misunderstanding the question? – Jojo Oct 16 '15 at 14:05
  • @GrahamKemp Or is it that HHT implies there is 1 consecutive Head and HHH means there are 2 and hence the answer is (1(1/8) + 1(1/8) + 2*(1/8))/3 ? – Jojo Oct 16 '15 at 14:13
  • @MJD Is my comment above accurate, from what you initially stated? – Jojo Oct 16 '15 at 14:14
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    you have $HHH$ happening with prob $\frac18$ and ${HHT, THH}$ happening with prob $\frac28$. Hence the expected value is $1\times \frac18+2\times\frac28=\frac58$. – Math-fun Oct 16 '15 at 14:17
  • @Math-fun Thanks. But MJD suggested looking at the number of consecutive heads in each outcome of throwing 3 coins, which seems slightly different from your answer. Sorry, I am slightly confused. – Jojo Oct 16 '15 at 14:52
  • @Math-fun If the "number of consecutive heads" is counting the number of "heads which follow immediately after another head", that should be $2\cdot\tfrac 1 8+ 1\cdot \tfrac 2 8 = \tfrac 4 8$. – Graham Kemp Oct 16 '15 at 14:59
  • @Jojo What is your definition of "number of consecutive heads"? For instance if HHH has three consecutive heads, and HHT has two consecutive heads, then does HTT have one consecutive head? or none? What about HTH? – Graham Kemp Oct 16 '15 at 15:01
  • @GrahamKemp Honestly, this is a question I found in a probability book that I am practicing and doesnt have answers, so I dont know what the correct definition is. But I feel that HHT has 1 and HHH has 2. – Jojo Oct 16 '15 at 15:02
  • @GrahamKemp HTT and HTH do not have consecutive heads. – Jojo Oct 16 '15 at 15:10

3 Answers3

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HHT THH :: 2/8 consecutive 2 heads

HHH :: 1/8 consecutive 3 heads

Samir
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The possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, and TTT. Of those, HHH, HHT, and THH have "consecutive heads". Now, do you mean "the number of times you get consecutive heads" or "the number of heads in flips that have consecutive heads"? In the first case, there are 3 cases with consecutive heads so the expected number of "consecutive heads" is 3/8. In the second case, there are a total of 3+ 2+ 2= 7 heads so the expected number is 7/8.

user247327
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Honestly, this is a question I found in a probability book that I am practicing and doesnt have answers, so I dont know what the correct definition is. But I feel that HHT has 1 and HHH has 2.

Well, then if the "number of consecutive heads" is counting the number of "heads which follow immediately after another head", then the expected number of consecutive heads will be : $$0\cdot \tfrac 5 8 +1\cdot\tfrac 28+ 2\cdot\tfrac 18 =\tfrac 48$$

Because of the eight equiprobable outcomes, five have no consecutive heads, two have one consecutive heads, and one has two consecutive heads.

Graham Kemp
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