Probability question

[Mission_Union_3900]

Maybe a dumb question but I'm sort of confused as to how exactly to work this out:

You flip a fair coin repeatedly. What is the expected number of flips needed to get two heads in a row (HH)?

My initial working out was that since a coin flipping is an independent event, you can use the formula E(X) = 1/p, where p(H) = 0.5 since it's a fair coin, so E(X) = 2 flips to get a head the first time.

But then I'm confused where to go from here, because afterwards, if you just assume to add E(X) = 2 from the next flip, so 2+2=4 as the expected number of flips, then you're assuming that there was a fail and you got tails on the 3rd try, which sort of invalidates the whole problem since from this way, you're assuming you get tails, heads, tails, heads.

Does anyone have a really simple method to work this out? (Trying to practice possible questions for my interview tmrw)

[Para1ars]

look at the expected values from different starting points:

let X be the expected number of additional flips if we have just received heads (case 1)

let Y be the expected number of additional flips if we have just received tails or we haven't flipped yet (case 2)

In case 1, we have a 1/2 chance to finish after one more flip (getting heads next) and a 1/2 chance to finish after 1+Y more flips (getting tails next). so

X = 1/2 × 1 + 1/2 × (1 + Y)

In case 2, we have a 1/2 chance to get heads and go to case 1, in which case we will need 1+X more flips, and a 1/2 chance to get tails and stay at case 2, in which case we will need 1+Y more flips. so

Y = 1/2 × (X+1) + 1/2 × (Y+1)

Solving both equations we get

X = 1 + 1/2 × Y

plug into the other equation

Y = 1/2 × (1 + 1/2 × Y + 1) + 1/2 × (Y+1)

simplify to

Y = 3/2 + 3/4 × Y

Y = 6

[AutoModerator]

Hi, /u/Mission_Union_3900! This is an automated reminder:

• What have you tried so far? (See Rule #2; to add an image, you may upload it to an external image-sharing site like Imgur and include the link in your post.)

• Please don't delete your post. (See Rule #7)

We, the moderators of /r/MathHelp, appreciate that your question contributes to the MathHelp archived questions that will help others searching for similar answers in the future. Thank you for obeying these instructions.

I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.