If the independent random variables X,Y are Binomially distributed with n = 3, p = and n = 5, p = respectively, then the probability of (X + Y ≥ 1) is:

Question

Question

This question was previously asked in

SSC CGL JSO Tier-II, 2018 (Statistics) Official Paper-III (Held On: 14 Sept, 2019)

Answer 1

Given

If the independent random variables X,Y are Binomially distributed that means

X ~ B(n, p)

X ~ B(3, 1/3) and Y ~ B(5, 1/3)

Here, B = Binomial distribution

n = number of observation

p = probability

Calculation

Since X and Y are independent binomial random variables with p₁ = p₂ = 1/3

By adding property of Binomial distribution

⇒ X + Y ~ B(3 + 5, 1/3)

⇒ X + Y ~B(8,1/3)

We know that, P(X + Y ≥ γ) = ⁸C_r(1/3)^r(2/3)^{8 - r}

⇒ P(X + Y ≥ 1) = 1 - P( X + Y < 1)

⇒ 1 - P(X + Y = 0)

∴ 1 - (2/3)⁸

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