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Suppose you flip a fair coin twice. What is the probability of getting two heads in a row (HH)? What about the probability of getting heads followed by tails (HT)?

Are the necessarily the same?

I thought they are necessarily the same because the probability of getting head or tails in a given toss is $\frac{1}{2}$.

So, $\text{P(HH})= \frac{1}{2} \times \frac{1}{2}$

And the same for P(HT)?

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  • $\begingroup$ You are correct. They are the same. $\endgroup$ – Remy Oct 29 '17 at 4:33
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    $\begingroup$ Notice that the question "What is the probability of getting heads, given you got heads last throw" which is also the set {H,H} is actually not the same as asking the probability of flipping two heads in a row. I know this isn't what you asked, but it's something to note if you're starting to learn Probability Theory. $\endgroup$ – Anthony P Oct 29 '17 at 4:59
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They are the same. The probability of getting heads is $.5$, and the probably of getting tails is $.5$. The two tosses are independent, so the probability of getting heads and then tails is $.5^2$, and so is the probability of getting heads and then heads.

A way to check is if the sum of the probabilities of all the possibilities of flipping a coin twice sums to $1$. $P(HH)=P(HT)=P(TH)=P(TT)=.5^2=.25$ and $4(.25)=1$.

Note that this can generalized. The probability of getting a sequence $HTTHHTHT$ is the same as getting $HHHHHHHH$.

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Yes. You can think of it as having two coins and flipping the both at the same time. The probability of each outcome HH HT HT TT. There are four possible outcomes, each having equal probability.

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Yes, the possibilities are (HH), (HT), (TH), and (TT). 1 possibility out of four is $1/4$.

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