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Answer

↓Steps Explained:↓

A binomial distribution has a probability of success = 0.4

Calculate the probability of you having at least 3 successes in 10 trials:

Binomial probability formula

f(k;n,p)  =	n! * pkqn - k
	k!(n - k)!

P(x >= 3) = 1 - P(x < 3) ΣP(x = k) where (0 <= k <= 3)

Calculate q:

q = 1 - p (q represents the probability of failure)

q = 1 - 0.4

q = 0.6

Calculate n!:

n! = 10!

10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

10! = 3628800

Calculate P(x = 0)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 0!

0! = 1

Calculate (n - k)!:

(n - k)! = (10 - 0)!

(n - k)! = 10!

10! = 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

10! = 3628800

Take our pieces and calculate the binomial probability:

P(x = 0)  =	10! * 0.400.6(10 - 0)
	0!(10 - 0)!

P(x = 0)  =	3628800 * 1 * 0.610
	1 * 3628800

P(x = 0)  =	3628800 * 1 * 0.0060466176
	3628800

P(x = 0)  =	21941.96594688
	3628800

P(x = 0) = 0.006

Calculate P(x = 1)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 1!

1! = 1

Calculate (n - k)!:

(n - k)! = (10 - 1)!

(n - k)! = 9!

9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

9! = 362880

Take our pieces and calculate the binomial probability:

P(x = 1)  =	10! * 0.410.6(10 - 1)
	1!(10 - 1)!

P(x = 1)  =	3628800 * 0.4 * 0.69
	1 * 362880

P(x = 1)  =	3628800 * 0.4 * 0.010077696
	362880

P(x = 1)  =	14627.97729792
	362880

P(x = 1) = 0.0403

Calculate P(x = 2)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 2!

2! = 2 * 1

2! = 2

Calculate (n - k)!:

(n - k)! = (10 - 2)!

(n - k)! = 8!

8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1

8! = 40320

Take our pieces and calculate the binomial probability:

P(x = 2)  =	10! * 0.420.6(10 - 2)
	2!(10 - 2)!

P(x = 2)  =	3628800 * 0.16 * 0.68
	2 * 40320

P(x = 2)  =	3628800 * 0.16 * 0.01679616
	80640

P(x = 2)  =	9751.98486528
	80640

P(x = 2) = 0.1209

Calculate P(x = 3)

Set x = 0 for the binomial probability formula

Calculate k!:

k! = 3!

3! = 3 * 2 * 1

3! = 6

Calculate (n - k)!:

(n - k)! = (10 - 3)!

(n - k)! = 7!

7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

7! = 5040

Take our pieces and calculate the binomial probability:

P(x = 3)  =	10! * 0.430.6(10 - 3)
	3!(10 - 3)!

P(x = 3)  =	3628800 * 0.064 * 0.67
	6 * 5040

P(x = 3)  =	3628800 * 0.064 * 0.0279936
	30240

P(x = 3)  =	6501.32324352
	30240

P(x = 3) = 0.215

Calculate cumulative probability

P(x > 3) = 1 - (P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3))

P(x > 3) = 1 - (0.006 + 0.0403 + 0.1209 + 0.215)

P(x > 3) = 1 - 0.3822

Calculate nq to see if we can use the Normal Approximation:

Since q = 1 - p, we have n(1 - p) = 10(1 - 0.4)

nq = 10(0.6)

nq = 6

Calculate the mean μ (expected value)

μ  =  np

μ  =  10 x 0.4

μ = 4

Calculate the variance σ²

σ²  =  np(1 - p)

σ²  =  10 x 0.4 x (1 - 0.4)

σ²  =  4 x 0.6

σ² = 2.4

Calculate the standard deviation σ

σ  =  √σ² = √np(1 - p)

σ  =  √2.4

σ = 1.5492

Calculate Skewness:

Skewness = 0.083333333333333

Calculate Kurtosis:

Kurtosis = -0.18333333333333

Final Answer