Let p be the probability of happening of an event.
Given p = 0.002 = \(\frac { 2 }{ 1000 } \)
Also n = 1000
∴ Mean = np = \(1000\times \frac { 2 }{ 1000 } \) = 2
Hence, X follows Poisson distribution with
P(x,λ) = \(\frac { e^{ -\lambda }.{ \lambda }^{ 0 } }{ x! } \)
∴ P( event happens exactly twice)
= P(X = 2)
= \(\frac { e^{ -\lambda }.{ \lambda }^{ 2 } }{ 2! } =\frac { e^{ -2 }.{ (2 }^{ 2 }) }{ 2 } \)
= e^-2(2) = 2(0.1353) = 0.2706
∴ P(X = 2) = 0.2706

Let p be the probability of winning the prize
Given p = \(\\ \frac { 1 }{ 100 } \) and n = 50
∴ Mean = np = \(\frac { 1 }{ 100 } \times 50=\frac { 1 }{ 2 } \)
∴ λ = 0.5
Hence, X follows Poisson distribution with
P(x,λ) = \(\frac { e^{ -\lambda }.{ \lambda }^{ 0 } }{ x! } \)
∴ P(winning the prize at least once) = P(X ≥ 1)
= 1-P(X < 1)
= 1-P(X = 0)
= 1-\(\frac { e^{ -\lambda }.{ \lambda }^{ 0 } }{ x! } \) = 1-e^-0.5
= 1-0.6066
∴ P(X≥1) = 0.3934

The probability function for the normal distribution is
f(x) = \(\frac { 1 }{ \sigma \sqrt { 2\pi } } e^{ -\frac { 1 }{ 2 } \left( \frac { x-\mu }{ \sigma } \right) ^{ 2 } }\), -∞ Given f(x) = \(C{ e }^{ -\left( \frac { x-100 }{ 50 } \right) ^{ 2 } }\)
= \(C{ e }^{ -\frac { 1 }{ 2 } \left( \frac { x-100 }{ 25 } \right) ^{ 2 } }\)
= \(C{ e }^{ -\frac { 1 }{ 2 } \left( \frac { x-100 }{ 5 } \right) ^{ 2 } }\) ...(2)
Comparing (1) and (2), μ =100, σ = 5 and
C = \(\frac { 1 }{ \sigma \sqrt { 2\pi } } =\frac { 1 }{ 5\sqrt { 2\pi } } \)
∴ C = \(\frac { 1 }{ 5\sqrt { 2\pi } } \).

Given n = 10
Probability of selecting a defective pen = p
= \(\frac { 10 }{ 50 } =\frac { 1 }{ 5 } \)
q = 1-p = \(1-\frac { 1 }{ 5 } =\frac { 4 }{ 5 } \)
∴ P(X = x) = ⁿC_x p^xq^n-x
P (at least one pen is defective)
= P(X≥1) = 1-P(X<1)
= 1-P(X = 0)
= 1-10C₀ \(\left( \frac { 1 }{ 5 } \right) ^{ 0 }\left( \frac { 4 }{ 5 } \right) ^{ 10 }\)
= 1-\(\frac { { 4 }^{ 10 } }{ { 5 }^{ 10 } } \)

Given mean μ = 60 and S.D. σ = 5

To find P(X > 60)
When X = 60, Z =\(\frac { X-\mu }{ \sigma } =\frac { 60-60 }{ 5 } \) = 0
∴ P(X > 60) = P(Z > 0) = P (0 < Z < ∞)
= 0.5
∴ 50% of students scored more than 60 marks