# Find the probability distribution of the random variable X.

A fair coin is flipped $3$ times. Consider a random variable $X$ which is the number of runs. Number of runs is the number of changes of letter $H$ and $T$. For example, $HHH$ has one run, $TTH$ has two runs and $THT$ has three runs. Find the probability distribution of the random variable $X$.

My work: I don't understand the phrasing of this question. In examples in my textbook and online $X$ is defined as the number of heads or tails. But I can't follow where the example in this question is going. I would think that $TTH$ and $THT$ would both have 2 runs since $HHH$ only has one. I don't know what zero runs would be either. Can anyone give me guidance on what exactly this question means? I'm pretty sure I can solve it once I understand what the number of runs means.

The outcomes would be:

$HHH$ $X=1$

$HTH$

$HHT$

$THH$

$TTH$ $X=2$

$HTT$

$THT$ $X=3$

$TTT$

I don't know what number of $X$ would correspond with each.

• Why should you think that there must be an outcome described by the event zero runs? In my interpretation, $\text{Onerun}=\{HHH,TTT\}, \text{Tworuns}=\{HHT,HTT,TTH,THH\}, \text{Threeruns}=\{HTH,THT\}$. Something like $TTHHHHTTHTHHHT$ would have $7$ runs. $\underbrace{TT}_1 \underbrace{HHHH}_2 \underbrace{TT}_3 \underbrace{H}_4 \underbrace{T}_5 \underbrace{HHH}_6 \underbrace{T}_7$ – JMoravitz Feb 28 '16 at 1:13
• Thank you that makes sense to me! – Koalafications Feb 28 '16 at 1:20

Thus $P(X=1) = 2/8, P(X=2) = 4/8, P(X=3) = 2/8$ is the distribution of $X$.