Question

- Assume that when adults with smartphones are randomly selected, 65% use them in meetings or classes. If 5 adult smartphone users are randomly selected, find the probability that exactly 3 of them use their smartphones in meetings or classes.

- Assume that when adults with smartphones are randomly selected, 41% use them in meetings or classes. If 20 adult smartphone users are randomly selected, find the probability that exactly 13 of them use their smartphones in meetings or classes.

- Assume that when adults with smartphones are randomly selected, 48% use them in meetings or classes. If 6 adult smartphone users are randomly selected, find the probability that at least 2 of them use their smartphones in meetings or classes.

- Assume that when adults with smartphones are randomly selected, 49% use them in meetings or classes. If 12 adult smartphone users are randomly selected, find the probability that fewer than 4 of them use their smartphones in meetings or classes.

Answer 1

Solution:

X follows Binomial(n , p) , then the PMF of X is

P(X = x) = (_n C _x) * p^x * (1 - p)^{n - x} ; x = 0 ,1 , 2 , ....., n

1)

p = 65% = 0.65

n = 5

So ,

P(Exactly 3)

= P(X = 3)

=  (₅ C ₃) * 0.65³ * (1 - 0.65)^{5 - 3}

= (₅ C ₃) * 0.65³ * (0.35)²

=  0.336415625

P(Exactly 3) = 0.336415625

2)

p = 41% = 0.41

n = 20

So ,

P(Exactly 13)

= P(X = 13)

=  (₂₀ C ₁₃) * 0.41¹³ * (1 - 0.41)^{20 - 13}

=  0.01784711339

P(Exactly 13) = 0.01784711339

3)

p = 48% = 0.48

n = 6

P(At least 2)

= P(X 2)

= 1 - { P(X < 2) }

= 1 - { P(X = 0) + P(X = 1) }

= 1 - { (₆ C ₀) * 0.48⁰ * (1 - 0.48)^{6 - 0} +  (₆ C ₁) * 0.48¹ * (1 - 0.48)^{6 - 1} }

= 1 - { 0.01977060966 + 0.10949876122 }

=  0.87073062912

P(At least 2) =  0.87073062912

4)

p = 49% = 0.49

n = 12

P(fewer than 4)

= P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

=  (₁₂ C ₀) * 0.49⁰ * (1 - 0.49)^{12 - 0} +  (₁₂ C ₁) * 0.49¹ * (1 - 0.49)^{12 -1} +  (₁₂ C ₂) * 0.49² * (1 -0.49)^{12 - 2}+  (₁₂ C ₃) * 0.49³ * (1 - 0.49)^{12 - 3}

= 0.00030962934 + 0.00356984421 + 0.01886417673 + 0.06041468366

= 0.08315833394

P(fewer than 4) = 0.08315833394