What is the probability that a psychic correctly "predicts" the outcome of at least 5 out of 10 coin flips? Assume the psychic is actually just randomly guessing on each flip.

The attempt:


*

*let E be the event in question

*number of outcomes per flip = 2

*chance of correctly guessing the correct outcome = $ \frac{1}{2}$

*notice the key word "at least": indicates that we should be doing a summation

*You can get exactly 5 correct, exactly 6, ..., all the way to exactly 10 correct
$$ P(E) = \sum_{i=5}^{10} (\frac{1}{2^i})= 63/1024$$

What is wrong with this approach?
My intuition says that the probability should be $ \frac{1}{2} $.
 A: The probability for each sequence of coin flips is $\frac{1}{2^{10}}$. 
There are $\binom{10}{k}$ sequences in which exactly $k$ predictions are correct (because there are $\binom{10}{k}$ ways to select the $k$ flips which are different from the psychic's guess).
Therefore you want $\sum\limits_{i=5}^{10}\binom{10}{i}/2^{10}$.
Using symmetry of the binomial coefficient this is equal to $\frac{2^{10}+\binom{10}{5}}{2}/2^{10}=\frac{1}{2}+\binom{10}{5}/2^{11}=\frac{1}{2}+\frac{252}{2048}\approx0.62$
A: The probability of guessing exactly 5 results is $\dbinom{10}{5}\dfrac 1 {2^{10}}$.  That is $\dfrac {63}{216}$.
The probability of guessing more than 5 results is $\dfrac{193}{512}$ and of guessing less than 5 results is $\dfrac{193}{512}$
Thus the probability of the 'psychic' guessing at least 5 of 10 coins is $\dfrac{319}{512}$.  
That is the confidence trick.   By including the not too unlikely possibility of "exactly 5", the 'psychic' gains an advantage over your intuition's estimate of $\tfrac 12$.
