Question

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

Answer 1

`\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}`

Answer 2

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`