We flip a biased coin 100 times.
H: Coin lands on the head
T: Coin lands on the tail
Let $P(H)=\frac{2}{3} $ and $P(T)=\frac{1}{3}$.
Suppose that we win 1 dollar if the coin lands on its head, and lose 1 dollar if it lands on its tail.
What is the probability that we will end up with at least $50?
I am trying to develop my intuition about these kinds of problems since I was asked on the spot and could not produce an answer in a timely manner. I went on to calculate the expected value which would in this case be ~33 dollars (~66 heads, ~33 tails), and was stuck afterwards. We clearly want $P(H\geq75)$ or $P(T\leq25)$.
How would you formulate it properly? What distribution do I use? How to obtain some intuition about these problems?
Best Answer
Your calculation of the expected value is a good start. The next step is to use the normal approximation to compute the standard deviation of the number of heads, which is $\sqrt{Np(1-p)}$. For $N=100, p=\frac 23,$ this is about $4.71$. You need $8 \frac 13$ extra heads over the expectation, which is $1.76 \sigma$ high. You can look up in the z-score table that you only beat that about $4\%$ of the time.