<Dl> <Dd> Pr (⋂ i = 1 n A i) = ∏ i = 1 n Pr (A i) = 1 2 n (\ displaystyle \ Pr \ left (\ bigcap _ (i = 1) ^ (n) A_ (i) \ right) = \ prod _ (i = 1) ^ (n) \ Pr (A_ (i)) = (1 \ over 2 ^ (n))). </Dd> </Dl> <Dd> Pr (⋂ i = 1 n A i) = ∏ i = 1 n Pr (A i) = 1 2 n (\ displaystyle \ Pr \ left (\ bigcap _ (i = 1) ^ (n) A_ (i) \ right) = \ prod _ (i = 1) ^ (n) \ Pr (A_ (i)) = (1 \ over 2 ^ (n))). </Dd> <P> Now suppose that we have just tossed four heads in a row, so that if the next coin toss were also to come up heads, it would complete a run of five successive heads . Since the probability of a run of five successive heads is only 1 / 32 (one in thirty - two), a person subject to the gambler's fallacy might believe that this next flip was less likely to be heads than to be tails . However, this is not correct, and is a manifestation of the gambler's fallacy; the event of 5 heads in a row and the event of "first 4 heads, then a tails" are equally likely, each having probability 1 / 32 . Given that the first four tosses turn up heads, the probability that the next toss is a head is in fact, </P> <Dl> <Dd> Pr (A 5 A 1 ∩ A 2 ∩ A 3 ∩ A 4) = Pr (A 5) = 1 2 (\ displaystyle \ Pr \ left (A_ (5) A_ (1) \ cap A_ (2) \ cap A_ (3) \ cap A_ (4) \ right) = \ Pr \ left (A_ (5) \ right) = (\ frac (1) (2))). </Dd> </Dl>

Probability of flipping heads 5 times in a row