# Flipping 2 coins, why is it more likely to get 1 head, 1 tail instead of 2 tails or 2 heads?

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The reason why it's more likely to get one head and one tail (than to get two heads or two tails) is analogous to the reason it's more likely to get ten heads and ten tails, when you flip twenty coins, than it is to get twenty heads or twenty tails. – Brian Tung Jun 10 '15 at 18:47

When flipping two coins, there are four possibilities:

$$\begin{array}{c|c} \color{blue}{\mathsf{\text{first coin = H, second coin = H}}}\strut & \color{red}{\mathsf{\text{first coin = T, second coin = H}}}\\\hline \color{red}{\mathsf{\text{first coin = H, second coin = T}}}\strut & \color{green}{\mathsf{\text{first coin = T, second coin = T}}}\\ \end{array}$$ There are two ways of getting one head and one tail (marked in red), whereas there is one way of getting two heads (marked in blue), and one way of getting two tails (marked in green). Because each of the four possibilities is equally likely, that means it is

• twice as likely to get (one head and one tail) than it is to get (two heads)
• twice as likely to get (one head and one tail) than it is to get (two tails)
• equally likely to get (one head and one tail) and (two heads or two tails)
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There are exactly four possible outcomes when flipping two coins, each of the four ways is equally likely. These are:

1. H H
2. H T <----
3. T H <----
4. T T

There are two ways to get one head and one tail.

There is only one way to get two heads.

There is only one way to get two tails.

Hence you have a 50% chance you'll get one head and one tail. You have only 25% chance you will you get all heads, and only 25% chance of getting two tails. So you are twice as likely to get one of each than you are of getting two heads, and likewise with respect to getting two tails.

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Gets a upvote from me! +1 – Amzoti Jun 4 '13 at 1:22

First of all, I think it is clear by symmetry that two heads are just as likely as two tails.

Now what about one of each? Imagine that you are tossing the coins one at a time. (Or, if you don't like that, imagine that you are tossing a $10$ cent coin and a $25$ cent coin.)

Whatever you got on the first toss (or on the $10$ cent coin), the probability that the second toss (or the $25$ cent coin) gives you a different result is $\frac{1}{2}$. This is because the second coin doesn't know what the result on the first coin was, or if it does know, it doesn't care.

So the probability you get a "mixed" result is $\frac{1}{2}$, and therefore the probability you get an "unmixed" result is also $\frac{1}{2}$. An unmixed result is of two types, two heads or two tails. Each, as we saw, is equally likely, so two heads has probability one-half of $\frac{1}{2}$, that is, $\frac{1}{4}$. Similarly, two tails has probability $\frac{1}{4}$. The mixed result has probability $\frac{1}{2}$. So the probabilities of two heads, two tails, and mixed are not all equal.

Remarks: $1.$ Note that the probability of getting head, then tail is the same as the probability of getting head, then head. They are each $\frac{1}{4}$. Equivalently, the probability of getting head on the $10$ cent coin, and tail on the $25$ cent coin, is $\frac{1}{4}$. But we also get a "mixed" result if we get a tail on the $10$ cent coin and a head on the $25$ cent coin. Thus the probability of a mixed result is $\frac{1}{4}+\frac{1}{4}$.

$2.$ Remembering that coins have no memory can be useful. Many people falsely believe that if, for example, we got three tails in a row, then head is, in some sense, "overdue," and has become more likely. That is not true. If one meets a person with such false beliefs, one might as well take advantage and make some money. If the person resents having been taken advantage of, call it a tuition fee.

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+1 for taking advantage of people who don't know math. en.wikipedia.org/wiki/Gambler%27s_fallacy is relevant. – Zen Jun 4 '13 at 1:10

There are two posibilities to get head-tail:

1. The first coin gives head, the second coin gives tail
2. The first coin gives tail, the second coin gives head

1. The first coin gives head, and so does the second.

There's only one possibility to get tail-tail:

1. The first coin gives tail, and so does the second.