I flip a biased coin, p = 0.5 for getting heads. What is the probability of getting heads 8 times ? Firstly I used probability intersection $$ P(A \cap B \cap C \cap D \cap E \cap F \cap G \cap H) = 0.5*0.5*0.5*0.5*0.5*0.5*0.5*0.5 $$ $$ = {1\above 1pt 256} $$
Now If i try with the Binomial distribution, I will get exactly the same result. My question is, why should I use binomial distribution if the probability intersection is enough ?
Thank you for your explanation.
Binomial distribution is more general concept that can also be used in other cases. For example you can ask: "What is the probability of getting 5 heads and 3 tails in 8 flips?"
In your example you can use probability of intersection or binomial distribution. Both are correct in this example.
However be careful. Formula $$P(A_1 \cap A_2 \cap \ldots \cap A_n)=P(A_1) \cdot P(A_2) \cdots P(A_n)$$ is satisfied only for mutually independent events.