Probability / permutations?

Question

A coin is tossed 6 times. Find the probability of getting exactly 3 heads

Answer 1

With 6 tosses there are 64 possible outcomes. (2*2*2*2*2*2)
There is 1 possibility for no heads.
There is 1 possibity for 6 heads.
There are 6 possibilities for 1 head/5 tails.
There are 6 possibilities for 5 heads/1tail.
There are 15 possibilities for 2 heads/4 tails.
There are 15 possibilities for 4 heads/2 tails.
There are 20 possibilities for 3 heads/3 tails.

1 + 1 + 6 + 6 + 15 + 15 +20 = 64

The 20 possible combinations of 3 heads are
1-HHHTTT
2-HHTHTT
3-HHTTHT
4-HHTTTH
5-HTHHTT
6-HTHTHT
7-HTHTTH
8-HTTHTH
9-HTTHHT
10-HTTTHH
11-THHHTT
12-THHTHT
13-THHTTH
14-THTHHT
15-THTHTH
16-THTTHH
17-TTHHHT
18-TTHHTH
19-TTHTHH
20-TTTHHH

So the probablity is 20 out of 64, which can also be stated as 5 to 16 or as 1 to 3.2.

Answer 2

P(x = 6) = 6C3 *(1/2)^3*(1/2)^(6-3)

The 6C3 is the combination formula. I can't work it out right now because my calculator is dead. The 1/2's are because the probability of getting heads is 1/2 and the probability of getting tails is 1/2.

In general, the formula is nCr * p^r * (1-p)^(n-r)
where n is the number of coins, r is the number of "successes" (heads in this case), p is the probability of success.