Question

A fair coin is tossed n times. What is the probability that no two consecutive heads appear? A fair coin is tossed n times. What is the probability that no two consecutive heads appear? @Mathematics

Answer 1

2^(n+1) - 2
i think

Answer 2

Let f(n) be the number of sequences of heads and tails, of length n, in which two consecutive heads do not appear.
The total number of possible sequences from n coin tosses is 2n.
So the probability that no two consecutive heads occur in n coin tosses is f(n) / 2n.

By enumeration, f(1) = 2, since we have {H, T}, and f(2) = 3, from {HT, TH, TT}.

We then derive a recurrence relation for f(n), as follows.

A sequence of n > 2 coin tosses has no consecutive heads if, and only if:

It begins with a tail, and is followed by n−1 tosses with no consecutive heads; or
It begins with a head, then a tail, and is followed by n−2 tosses with no consecutive heads.

These two possibilities are mutually exclusive, so we have f(n) = f(n−1) + f(n−2).

This is simply the Fibonacci sequence, shifted forward by two terms.

The Fibonacci sequence is defined by the recurrence equation F1 = 1, F2 = 1, Fk = Fk−1 + Fk−2, for k > 2.
So F3 = 2 and F4 = 3, and therefore f(n) = Fn+2.

A closed form formula for the Fibonacci sequence is Fn = (Phin − phin) /root 5,
where Phi = (1 + root 5)/2 and phi = (1 − root 5)/2 are the roots of the quadratic equation x2 − x − 1 = 0.

Therefore the probability that no two consecutive heads appear in n tosses of a coin is Fn+2 / 2n = (Phin+2 − phin+2) / 2n·root 5.

Answer 3

The probability that no two consecutive heads appear in n tosses of a coin is Fn+2 / 2n,
where Fn is the Fibonacci sequence, defined by the recurrence relation Fn = Fn−1 + Fn−2, for n > 2, with F1 = F2 = 1.

A closed form formula for the Fibonacci sequence is Fn = (Phin − phin)/root 5,
where Phi = (1 + root 5)/2 and phi = (1 − root 5)/2 are the roots of the quadratic equation x^2 − x − 1 = 0.

Therefore the above probability can also be written as (Phi^n+2 − phi^n+2) / 2n·root 5.

Answer 4

i feel it will be always 1/4

Answer 5

1/4 is the probability of two consecutive heads so for no two consecutive two heads it will be 3/4