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When a certain coin is flipped, the probability of heads is $0.5$. If the coin is flipped $6$ times, what is the probability that there are exactly $3$ heads?

The answer is $\frac5{16}$. I wonder why it isn't $\frac12$. Since a fair coin flip results in equally likely outcomes, any sequence is equally likely…

I know why it is $\frac5{16}$. We divide the number of possible outcomes with exactly 3 heads by the total possible outcomes.

What bothers me is how I should think about it so I won't make a mistake anymore. Why is $\frac12$ not right? I need to get intuition.

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    $\begingroup$ Above all, though, I would advise you to do more probability problems. You learn mathematics by doing mathematics. $\endgroup$ – Parcly Taxel Oct 29 '16 at 4:50
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    $\begingroup$ Suppose you flip the coin $5$ times. What is the probability that there are exactly $2.5$ heads? $\endgroup$ – Théophile Oct 29 '16 at 5:34
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It sounds like you already have the intuition since you understand that the answer is obtained by dividing the number of outcomes with exactly 3 heads by the total number of outcomes. From here it's a matter of understanding how to calculate these two things.

The total number of outcomes is simply $2^6 = 64$ since we're tossing a coin 6 times and each toss has only two possible outcomes.

The number of outcomes with exactly 3 heads is given by ${6 \choose 3}$ because we essentially want to know how many different ways we can take exactly 3 things from a total of 6 things. The value of this is 20.

So the answer is $20/64 = 5/16$.

The error you made is thinking that "number of outcomes with exactly 3 heads" is equal to "half of the total number of outcomes of 6 tosses." If this were the case then logically, "exactly 3 tails" must also be exactly half of the total outcomes. This means that "exactly 3 heads or exactly 3 tails" must describe all possible outcomes (because each scenario joined by the "or" would have probability $1/2$) but this is clearly not the case since we can have, e.g., 1 heads and 5 tails, etc.

To put it another way... You said any sequence is equally likely. That is correct. But sequences containing exactly 3 heads do not make up half of the total number of sequences. Therefore it does not follow that the probability is $1/2$. The easiest way to see this clearly is to list every possible outcome. But for 64 outcomes it can be tedious, so let's do it with a simpler and similar problem.

Say we want to know the probability of getting exactly 2 heads if we flip a coin 4 times. Unless I'm misunderstanding your misunderstanding, your earlier thinking would lead you to believe the answer is $1/2$. But if we list all possible outcomes:

HHHH HHHT HHTH HHTT HTHH HTHT HTTH HTTT THHH THHT THTH THTT TTHH TTHT TTTH TTTT

We see that only 6 of them have exactly 2 heads. $6/16 = 3/8$. And if we do this problem the way I answered the original one, then the total number of outcomes is $2^4 = 16$ and the total number of outcomes with exactly 2 heads is ${4 \choose 2} = 6$. So we again get $6/16 = 3/8$.

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"exactly three heads" is not half of the outcomes. Try writing the outcomes down and counting them!

The probability is $$P=\binom{6}{3}(\frac{1}{2})^3(\frac{1}{2})^{6-3}$$

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Your incorrect thinking goes along the lines of this classic fallacy:

What's the probability of this happening? $\frac12$, either it will happen or it won't.

It is true that each sequence of heads and tails is equally likely to occur – with probability $\frac1{64}$, in this case. However, the number of those sequences having exactly three heads is not 32, but $\binom63=20$, which leads to the correct answer of $\frac5{16}$. They are two completely different things.

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Since a fair coin flip results in equally likely outcomes, any sequence is equally likely…

... then we need to know how many of these equally-probable outcomes are possible and how many of which have "exactly three heads".

There are $2^6$ ways to make six independent "head or tails" decisions. That is $64$.

There are $\binom 6 3$ ways to arrange three heads and three tails. That is $20$

Divide and calculate. $20/64 = 5/16$

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