Probability question easy math!?

Question

A fair coin is tossed 11 times. What is the probability that the coin lands head at least 9 times?

a) 0.0322

b) 0.0654

c) 0.0317

d) 0.0327

e) 0.0269

Answer 1

Let's start with the easy case:

Probability coin lands heads exactly 11 times:
P(Heads = 11) = (0.5)^11

Now the other cases:

Probability coin lands heads exactly 10 times:
P(Heads = 10) = C(11,1) * (0.5)^11
Here you pick the ways to have one coin land tails and the rest land heads. There are 11 choices for the coin to land tails. (Note: you could also use C(11,10) which comes out the same value of 11)

Probability coin lands head exactly 9 times:
P(Heads = 9) = C(11,2) * (0.5)^11
Again you choose 2 coins to land tails. Order doesn't matter because picking 1st and 2nd tosses to be tails is the same as 2nd and 1st toss being tails. The formula is (11 * 10) / 2 = 55. (Again you could use C(11,9) which is the same thing.)

The final probability of having *at least* 9 heads is the sum of all the three probabilities above:

P( Heads ≥ 9):
= P( Heads = 9) + P( Heads = 10) + P( Heads = 11)
= [ C(11,2) + C(11,1) + C(11,0) ] 0.5^11
= (55 + 11 + 1) * (0.5)^11
≈ 0.0327148438
≈ 3.27%

The answer is therefore:
D) 0.0327