If you flip a fair coin 10 times, what is the probability that it lands on heads exactly 4 times?

2 Answers

#\frac{105}{2^9}#

Explanation:

The probability of getting a head in a single toss

#p=1/2#

The probability of not getting a head in a single toss

#q=1-1/2=1/2#

Now, using Binomial theorem of probability,

The probability of getting exactly #r=4# heads in total #n=10# tosses

#=^{10}C_4(1/2)^4(1/2)^{10-4}#

#=\frac{10\times9\times8\times7}{4!}\frac{1}{2^4}\frac{1}{2^6}#

#=\frac{2^4\cdot 9\cdot 35}{24(2^{10})}#

#=\frac{105}{2^9}#

Jun 26, 2018

The probability is approximately 20.51%.

Explanation:

This question uses the binomial distribution.

Let #X# be the number of heads in 10 tosses.
Then #X# is distributed as #"Bin"(n=10," "p=1/2)#.

The probability of #X# being 4 is therefore

#"P"(X=4)=""_10C_4(1/2)^4(1-1/2)^(10-4)#

#color(white)("P"(X=4))=210(1/16)(1/2)^6#

#color(white)("P"(X=4))=210(1/16)(1/64)#

#color(white)("P"(X=4))=210/1024=105/512~~0.2051#