Question

A coin is flipped seven times. What is the probability of getting heads six or fewer times?

I know how to solve this with the probability equation.

2x2x2x2x2x2x2=128 total outcomes

Which is p1+p2=1 -> p1=1-p2-=1-1/128= 127/128 is the answer.

But whenever I try to solve it with the permutation formula I get a wrong answer

n!/ k!(n!-k!) = 7!/6!(7!-6!) = 7 7/128 where am I wrong? can someone explain to me what am I doing wrong?

Answer 1

First of all, the permutation formula is wrong. It should be n!/ k!(n-k)! instead of n!/ k!(n!-k!).

By the permutation formula, n!/ k!(n!-k!) = 7!/6!(7-6)! = 7 is the number of possible cases of getting exactly six heads.

For calculating the probability of getting heads six or fewer times, it is needed to calculate the probability of getting exactly five heads, exactly four heads, exactly three heads, exactly two heads, exactly one head and exactly zero head and adding all these probabilities with the probability of getting exactly six heads to get the final required probability of getting six or fewer heads.

The number of possible cases of getting exactly six heads = n!/ k!(n-k)! = 7!/ 6!(7-6)! = 7
Probability of getting exactly five heads = 7/128

The number of possible cases of getting exactly five heads = n!/ k!(n-k)! = 7!/ 5!(7-5)! = 21
Probability of getting exactly five heads = 21/128

The number of possible cases of getting exactly four heads = n!/ k!(n-k)! = 7!/ 4!(7-4)! = 35
Probability of getting exactly five heads = 35/128

The number of possible cases of getting exactly three heads = n!/ k!(n-k)! = 7!/ 3!(7-3)! = 35
Probability of getting exactly five heads = 35/128

The number of possible cases of getting exactly two heads = n!/ k!(n-k)! = 7!/ 2!(7-2)! = 21
Probability of getting exactly five heads = 21/128

The number of possible cases of getting exactly one head = n!/ k!(n-k)! = 7!/ 1!(7-6)! = 7
Probability of getting exactly one head = 7/128

The number of possible cases of getting exactly zero head = n!/ k!(n-k)! = 7!/ 0!(7-0)! = 1
Probability of getting exactly zero head = 1/128

Thus the probability of getting heads six or fewer times
= (7/128) + (21/128) + (35/128) + (35/128) + (21/128) + (7/128) + (1/128) = 127/128