# Let #X# be a binomial random variable with #p=0.4# and #n=10#. What is # P(X = 5) #?

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The probability ( P(X = 5) ) for a binomial random variable ( X ) with parameters ( p = 0.4 ) and ( n = 10 ) can be calculated using the binomial probability formula:

[ P(X = k) = \binom{n}{k} \times p^k \times (1 - p)^{n - k} ]

Substitute ( k = 5 ), ( p = 0.4 ), and ( n = 10 ) into the formula:

[ P(X = 5) = \binom{10}{5} \times (0.4)^5 \times (0.6)^5 ]

Calculate the binomial coefficient ( \binom{10}{5} ) using the formula:

[ \binom{n}{k} = \frac{n!}{k!(n-k)!} ]

[ \binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252 ]

Plug in the values and compute:

[ P(X = 5) = 252 \times (0.4)^5 \times (0.6)^5 ]

[ P(X = 5) = 252 \times 0.01024 \times 0.07776 ]

[ P(X = 5) = 252 \times 0.0007969 ]

[ P(X = 5) \approx 0.20051 ]

So, ( P(X = 5) ) is approximately ( 0.20051 ) or ( 20.05% ).

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