Question: 2 heads

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Comment on 2 heads

You can also solve this simply by multiplying .9*.9*.1*.1*.1 to get 0.0081 since order doesn't matter.
greenlight-admin's picture

That's not quite correct.

Your equation only accounts for one of many possible cases. That is, your equation is for calculating the probability of Heads - Heads - Tails - Tails - Tails

You also need to consider other cases like:
- Tails - Heads - Heads - Tails - Tails
- Tails - Tails - Heads - Tails - Heads
- Tails - Tails - Tails - Heads - Heads
etc.

If you evaluate your equation (.9*.9*.1*.1*.1), you get 0.00081 (not 0.0081)

Q.N.: A fair coin is to be tossed 5 times. What is the probability that exactly 3 of the 5 tosses result in heads?

7⁄32
1⁄4
5⁄16
3⁄8
7⁄16
greenlight-admin's picture

Let's examine ONE case in which we get exactly 3 heads: HHHTT

P(HHHTT) = (1/2)(1/2)(1/2)(1/2)(1/2) = 1/32

This, of course, is just ONE possible way to get exactly 3 heads.

Another possible outcome is HHTTH

Here, P(HHTTH) = (1/2)(1/2)(1/2)(1/2)(1/2) = 1/32

As you might guess, each possible outcome will have the same probability (1/32). So, the question becomes "In how many different ways can we get exactly 3 heads and 2 tails?"

In other words, in how many different ways can we arrange the letters HHHTT?

Well, we can apply the MISSISSIPPI rule (from the counting module) to see that the number of arrangements = 5!/(3!)(2!) = 10

So P(exactly 3 heads) = (1/32)(10) = 10/32 = 5/16

Or we can also choose the binomial formula:

(n choose k)Xp^kX(1-p)^n-k
(5 choose 2)X0.9^2X0.1^3
10X0.81X0.001
0.0081
greenlight-admin's picture

That works too!

Thanks for the wonderful lessons of probability..they were very helpful and cleared many small doubts.
greenlight-admin's picture

Thanks Deepak!

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