Suppose X has a binomial distribution `B(6,""""1/2)` . Show that `X" "=" "3` is the most likely outcome. (Hint: `P(x=3)` is the maximum among all `P(x_i),""x_i=0,""1,""2,""3,""4,""5,""6)`

Explanation

`{X}` is the random variable whose binomial distribution is `B(6, frac{1}{2}) . .`

Therefore, `n=6 and p=1 / 2`

`therefore q=1-p=1-frac{1}{2}=frac{1}{2}`

Then, `{P}({X}=x)={ }^{n} {C}_{x} q^{n-x} p^{x}`

`=^{6} {C}_{x}(frac{1}{2})^{6-x} cdot(frac{1}{2})^{x} `

`=^{6} {C}_{x}(frac{1}{2})^{6}`

It can be seen that` P(X=x)` will be maximum, if `{ }^{6} {C}_{x}` will be maximum.

Then, `{ }^{6} {C}_{0}={ }^{6} {C}_{6}=frac{6 !}{0 ! cdot 6 !}=1`

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