# Probability of getting exactly 2 heads in 3 coins tossed with order not important?

I have been thinking of this problem for the post 3-4 hours, I have come up with this problem it is not a home work exercise

Let's say I have 3 coins and I toss them, Here order is not important

so possible sample space should be

0 H, 1 H, 2 HH, 3 HHH (H being heads) TTT, HTT, HHT, HHH

since P(T) and P(H) =1/2;

Here we have fair coins only, Since each and every outcome is equally likely, answer should be

1/4 (is this correct)

and if that is correct, all of the probabilities don't add up to one, will I have to do the manipulation to make it add up to one, or I am doing anything wrong.

EDIT In my opinion, with order being not important, there should be only 4 possible outcomes. All of the answers have ignored that condition.

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are you looking for the probability to get exactly 2 heads or at last 2 heads in 3 tosses? –  StringerBell Apr 17 '13 at 22:45
The events of $0$, $1$, $2$, and $3$ heads are not equally likely. –  Sammy Black Apr 17 '13 at 22:46
@StringerBell exactly 2 heads, updated the question –  Dude Apr 17 '13 at 22:48
@SammyBlack Can you please explain why? Both head and tall have got 1/2, so all of the setting should have 1/8. –  Dude Apr 17 '13 at 22:50
Do you feel like the following sufficiently captures your concept of order not important: "Let's say we flip three separate coins, and then arrange them so that the heads are first and then the tails are last. What is the probability that we end up with the sequence $\rm HHT$?" –  George V. Williams Apr 17 '13 at 23:07

The sample space has size $2^3 = 8$ and consists of triples $$\begin{array}{*{3}{c}} H&H&H \\ H&H&T \\ H&T&H \\ H&T&T \\ T&H&H \\ T&H&T \\ T&T&H \\ T&T&T \end{array}$$

The events \begin{align} \{ 0 \text{ heads} \} &= \{TTT\}, \\ \{ 1 \text{ head} \} &= \{HTT, THT, TTH\}, \end{align} and I'll let you figure out the other two.

The probabilities are, for example, $$P(\{ 1 \text{ head} \}) = \frac{3}{8}.$$

This is called a binomial distribution, and the sizes of the events "got $k$ heads out of $n$ coin flips" are called binomial coefficients.

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I know that, but (for some stupid reason) I have a put up a restriction here, order not being important. So HTH, HHT are same. –  Dude Apr 17 '13 at 22:55
You are saying "order is not important" but this regards counting the number of heads. Order is still important in terms of the different possibilities. –  Sammy Black Apr 17 '13 at 22:57
No I am saying 2 heads and 1 tail is 2 Heads and 1 tail, why am I being forced to calculate their combinations. May be I am being absurd, But I think it is a valid point. –  Dude Apr 17 '13 at 22:59
Try actually flipping these coins. Do 100 sets or so. You'll get about 3 times more "2 heads" than "3 heads". –  Henry Swanson Apr 17 '13 at 23:10

Consider all the possible ways to get two heads, $\rm HHT, HTH \; and \; THH$. There are $2 \cdot 2 \cdot 2 = 8$ possible combinations in total. Therefore, the answer is $3/8$.

Your answer is wrong because the number of ways of changing around $\rm HHT$ (3) is not the same as the number of ways of changing around $\rm HHH$ (1). Can you see why this would invalidate your argument?

General solution: Binomial distribution. The probability of getting $k$ successes (here $2$) in $n$ trials (here $3$) is given by:

$$\Pr(x=k) = \binom{n}{k} p^k (1-p)^{n-k}$$

Where $p$ is the probability of success (here, $p=1/2$), and $\binom{n}{k} = n!/(k!(n-k)!)$. This gives us:

$$\binom{3}{2} \left(\frac12\right)^2 \left(1 - \frac12\right)^{3-2}$$ $$3 \cdot \frac14 \cdot \frac12$$ $$\frac38$$

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The outcomes you are looking for are either THH, HTH or HHT. Taking a look at for example THH: the possibility to toss T or H is $0.5$. Thus the possibility to throw T and then H and then H is $\frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{8}$. But since we have three ways to "achieve" the desired result, the possibility of throwing exactly tow heads in three tosses is $\frac{1}{8} + \frac{1}{8} + \frac{1}{8} = \frac{3}{8}$.

If you continue you like this, you'll find that the possibilites of all possible outcomes (THH, TTH, TTT, HTT ...) add up to 1 indeed.

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Saying "order is not important" does not mean that each scenario (0, 1, 2, 3 heads) is equally likely. The probability of getting one head, for example, means that you got one head first, one head second, or one head third. There are 8 possible outcomes, so the probability of one head must be 3/8.

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