Say you flip a coin seven times. What is the probability the number of heads will be even?

Question

Say you flip a coin seven times. What is the probability the number of heads will be even?

Please attach detailed description of how one gets to answer

Answer 1

If you flip a coin 3 times there are 8 permutations.

Three of them will have an even number of heads and four of them will have an odd number of heads.

(plus TTT)

8-5=3

If you flip a coin 4 times there are 16 permutations.

Seven of them will have an even number of heads and eight of them will have an odd number of heads.

(plus TTTT)

16-9=7

If you flip a coin 7 times, there will be 128 permutations.

128-65=63

P(even number of heads)=63/128

take the number of flips, subtract (half of the flips+1) to get the answer (number of flips wanted)

Answer 2

You need the sum of 4 terms of the binomial density function, if you consider 0 heads even and 3 terms if you ignore 0 heads.

Even number of heads means 0, 2, 4, or 6 heads.

[₇C₀ + ₇C₂ + ₇C₄ + ₇C₆] * (1/2)⁷ where _nC_r is the binomial coefficient of n things taken r at a time.

If you ignore 0 heads, just omit the first term in the sum.