A coin is tossed 7 times. In how many ways can 3 consecutive heads be obtained?

Answer 1

5 ways

Answer 2

48

It might first appear that you can simply do 5 * 2^4 (note to thengine7, 2^4 = 16, not 8) and get 80, because you have HHH**** *HHH*** **HHH** ***HHH* and ****HHH. This is incorrect because by filling in all possible combinations of heads and tails for the *s you count both HHH(h)*** and (h)HHH***, even though they are the same.

How do we fix this? well, for every instance of HHHH, we want to remove 1 from our previous total of 80. By using the above method with four heads instead of 3 we get 4 * 2^3 = 32. There will be repeats here as well ( HHHH(h)** and (h)HHHH**) but this is ok, in fact, it saves us some time. In a group of 5 heads 4 heads will count twice, and 3 heads will count thrice. Thus subtracting our total will remove repeats for occurances of 4, and 5 heads in a row. If you try this with 6 and 7, you will see that it works there as well.

So, we take 80, remove the duplicates ( - 32) and are left with 48 occurances of 3 consecutive heads.

note: vishal, even though that isn't what the question asked, your answer is still wrong.

edit: mith, your stupidity doesn't make my answers "off the wall" HHHTTTT and HHHTHTT are two different ways that 3 consecutive heads can be obtained... understand yet genius?

Answer 3

5

Answer 4

The above poster is near correct, if you got HHH , you then have 8 different outcomes you don't care about, from 4 more tosses. 2^4 = 8
That also means that there are 4 more positions where you can have the 3 consecutive tosses happen. so thats 8 outcomes times 5 different positions = 40 different ways.
5 total positions.. HHHTTTT, THHHTTT, TTHHHTT, TTTHHHT, TTTTHHH. the tails can be heads or tails, we don't care.

Answer 5

ANS:5/128

Probability of getting a HEAD = 1/2
Probability of getting a TAIL = 1/2

Probability of getting 3 HEAD in 3 attempts =1/2 * 1/2 * 1/2
= 1/8

Probability of getting 3 HEAD in 4 attempts
=1/8 *1/2+ 1/2*1/8
=2 * ( 1/8 * 1/2 )
= 1/8
similarly....
Probability of getting three consecutive HEADS in 7 attempts
= 5 * (1/8 * 1/16)
= 5/128

Answer 6

I agree that their are 5 ways. I don't know what the others were doing. There off the wall answers made me spend alot of time trying to figure out what they were thinking (I never did).

Answer 7

Tosses
1,2,3
2,3,4
3,4,5
4,5,6
5,6,7
There are 5 ways