A fair coin is flipped ten times. What is the probability of the coin landing heads up exactly six times?

Question

A fair coin is flipped ten times. What is the probability of the coin landing heads up exactly six times?

Answer 1

 

This is a binomial probability, there is only two choices for outcome. (Success or not success), (True or False). Here (Head or Tail).

 

Every time we flip a coin it is a trial (here 10 trials) and all trials are independent of each other and random. 

 

 

probability for each trial:

H=heads, P(H)=1/2
T=tails, P(T)=1/2

 

P=p(H) and q=P(Not H)=P(T)

P+q=1 always in binomial probability.

 

One situation is (H H H H H H T T T T) that is 6 H and 4 T, we can have different combinations and the formula for combination of r out of n is:  _nC_r = n!/r!(n-r)!

 

General binomial formula: p(x= exact r times out of n trials) = _nC_r * p^r * q^(n-r)

 

Here since p=q=1/2 therefore the binomial formula looks simpler. 

_nC_r * (1/2)^r * (1/2)^(n-r) = _nC_r * (1/2)^{n = _nC_r / 2^n}

 

In this question, n=10, r=6, p=q=1/2 hence _nC_r=₁₀C₆=10!/(6!4!)=210

 

Therefore p(x= exact 6 tails) = 210/2^10 = 210/1024 = 105/512

 

Good Luck!