Calculate the probability of getting exactly 50 heads and 50 tails after flipping a fair coin 100 times. for this question we can easily apply the the binomial distribution formula, as $100 \choose 50$ $\frac{1}{2}^{100}$
What if we are asked
Calculate the probability of getting exactly consecutive 10 heads or 10 tails after flipping a fair coin 100 times? would binomial distribution still work. thank you
The probability of getting exactly consecutive 10 heads and 10 tails is $\frac{1}{2}^{20}$.
Then you have to consider that it can happen from zero to five times. So you have to use the Bernoulli’s Formula to calculate the total probability: $$\sum_{k=0}^{5} \binom{5}{k} \left( \frac{1}{2}^{20} \right)^{5} \left(1 - \frac{1}{2}^{20} \right)^{5-k}$$