Question
Suppose that we flip a fair coin 4 times.
(a) What is the probability of at least two consecutive heads are flipped given that the first flip is a heads?
(b) Are the events in the above part independent?
My attempt
(a) H will denote heads, and T denote tails. Since our first flip is already a H, there are $2^3=8$ different ways that the remaining flips can go. Of these, there are four ways we can get two or more consecutive heads: $HHHH$, $HTHH$, $HHTT$, and $HHTH$. I'm not sure if there is a more systematic way to approach this other than listing them out.
Since there are 4 ways that we can get two or more consecutive heads out of the 8 total, the probability is $\frac{4}{8}=\frac{1}{2}$.
(b) The two events are (A) a flip turns out heads (B) that there are because the coin is fair, and the probability of (B) I found by again listing out the ways we can get two or more consecutive heads $HHHH$, $HHHT$, $HHTT$, etc of a total $2^4=16$ possibilities two coins could take. There are 8 ways to get two consecutive heads. Again, I'm not sure if there's a more systematic approach here.
Two events are independent if their product is equal to the intersection of the events. $P(A)*P(B)= \frac{1}{2}*\frac{1}{2} = \frac{1}{4}$, which is not equal to the probability we found in part (a), $P(A\cap B) = \frac{1}{2}$. So the events are not independent.
Best Answer
Either you get another head in the second flip, or you get a tail and then two consecutive heads in the remaining flips. $\tfrac 12+\tfrac 18= \frac 58$
Which, are the outcomes: $\rm \mathbf HHTT, \mathbf HHTH, \bbox[wheat, 2pt]{\mathbf HHHT}, \mathbf HHHH$ and $\rm \mathbf HTHH$
The probability of getting heads in the first toss is $1/2$
The probability of getting two consecutive heads is $\binom 42 \tfrac 1{16}$
Even without calculating we can immediately see that the product will not be $5/8$
Intuitively: Because the head in the first flip can be part of two consecutive heads, we should anticipate a dependency; and indeed there are more outcomes possible for "having two consecutive heads" when the first toss is heads than when it is tails. (vis: Compare the above to : $\rm \mathbf THHT, \mathbf THHH$ and $\rm \mathbf TTHH$ .)