p-value calculation for coin flipping Here it mentioned we need to take into consideration of rare and rarer event (up to 12:28) for p-value calculation.
There is a problem:
You observed 3 sequences of 5 flips of coin with at least one sequence was all heads.
The hypothesis to test is - whether the coin is bias.
(The null hypothesis is the fair is not bias)
The probability calculation would be
 3 sequences of 5 flips of coin with at least one sequence was all heads
= prob of at least one sequence in 3 sequence was all heads  
= 1- prob of no all heads in 3 sequence 
= 1- (no all heads in 1 sequence)^3 
= 1 - (1-all heads in 1 sequence)^3 
= 1 - (1-(1/2)^5)^3 = 0.09

But what is the p-value?

It isn't well-defined.
The p-value is the probability of getting a result "at least as extreme" as the observed one, conditional on the coin being fair.
But we don't have full information about what the observed result was - we are only told that at least one sequence was all heads.
Suppose that all 5 sequences had each consisted of 4 heads and 1 tail - would that be "more extreme"? The question vaguely implies that the answer is no, and that we only care about how many sequences were all heads - but that's absurd, since it implies that HHHHH HHHHH TTTTT is "more extreme" than HHHHT HHHHT HHHHT, even though the latter result is obviously stronger evidence that the coin is biased towards heads.
