Probabilities (Math)
posted by Shane .
An experiment consists of tossing a coin 14 times.
(a) How many different outcomes are possible?
(b) How many different outcomes have exactly 9 heads?
(c) How many different outcomes have at least 2 heads?
(d) How many different outcomes have at most 10 heads?

a) each turn could be 2 ways, so 2^14 = 16384
b) this is the same as asking, "in how many ways can we arrange 9 H's and 5 T's, the H's and T's are indistinguishable.
number of ways = 14!/(9!5!) = 2002
c)at least two heads implies we don't want 0 heads, or 1 head, let's find those two
0 heads > 1 way
1 head > 14!/13! = 14
so at least 2 heads = 16384  14  1 = 16369
d) so we don't want 11 heads, 12 heads, 13 heads and 14 heads
which are 14!/(11!3!) + 14!/(12!2!) + 14!/(13!1!) + 1
= 364 + 91 + 14 + 1 = 470
so at most 10 heads = 16384  470 = 15914