Question

1. x is a binomial random variable. (Give your answers correct to three decimal places.)

(a) Calculate the probability of x for: n = 1, x = 0, p = 0.35 (b) Calculate the probability of x for: n = 7, x = 6, p = 0.7

(c) Calculate the probability of x for: n = 4, x = 1, p = 0.35

(d) Calculate the probability of x for: n = 3, x = 3, p = 0.7

(e) Calculate the probability of x for: n = 3, x = 0, p = 0.2

(f) Calculate the probability of x for: n = 6, x = 0, p = 0.7

Answer 1

Solution :

a ) Given that ,

p = 0.35

1 - p = 1 -0.35 = 0.65  

n = 1

Using binomial probability formula ,

P(X = x) = ((n! / (n - x)!) * p^x * (1 - p)^{n - x}

P(X = 0) = ((1! / (1 - 0)!) * 0.35⁰ * 0.65^{1 - 0}

=0.6500

Probability =0.6500

b )  Given that ,

p = 0.7

1 - p = 1 -0.7 = 0.3

n = 7

Using binomial probability formula ,

P(X = x) = ((n! / (n - x)!) * p^x * (1 - p)^{n - x}

P(X = 6) = ((7! / (7 - 6)!) * 0.7⁶ * 0.3^{7 - 6}

=0.2471

Probability =0.2471

c ) Given that ,

p = 0.35

1 - p = 1 -0.35 = 0.65  

n = 4

Using binomial probability formula ,

P(X = x) = ((n! / (n - x)!) * p^x * (1 - p)^{n - x}

P(X = 1) = ((4! / (4 - 1)!) 0.35¹* 0.65^{4 - 1}

=0.3845

Probability =0.3845:

d ) Given that ,

p = 0.7

1 - p = 1 -0.7 = 0.3  

n = 3

Using binomial probability formula ,

P(X = x) = ((n! / (n - x)!) * p^x * (1 - p)^{n - x}

P(X = 3) = ((3! / (3 - 3)!) * 0.7³ * 0.3^{3 - 3}

=0.3430

Probability =0.3430

e ) Given that ,

p = 0.2

1 - p = 1 -0.2 = 0.8  

n =3

Using binomial probability formula ,

P(X = x) = ((n! / (n - x)!) * p^x * (1 - p)^{n - x}

P(X = 0) = ((3! / (3- 0)!) * 0.2⁰ * (0.8^{3 -0}

=0.5120

Probability =0.5120

f ) Given that ,

p = 0.7

1 - p = 1 -0.7 = 0.3

n = 6

Using binomial probability formula ,

P(X = x) = ((n! / (n - x)!) * p^x * (1 - p)^{n - x}

P(X = 0) = ((6! / (6 - 0)!) * 0.7⁰* 0.3^{6 - 0}

=0.0007

Probability =0.0007