If the probability that any 50 year old person will die within a year

1.	If the probability that any 50 year old person will die within a year is 0.01, find the probability that out of a group of 5 such persons(i) none of them will die within a year(ii) at least one of them will die within a year.
Answer» p = Probability that any 50 year old person will die within a year = 0.01 ∴ q = 1 – p = 1 – 0.01 = 0.99 X is binomial random variable. ∴ Putting, n = 5, p = 0.01, q = 0.99 in P(X = x) = p(x) = ⁿC_xp^xq^n-x P(X = x) = p(x) = ⁵C_x(0.01)^x(0.99)^n-x (i) None of the 50 year old person will die within a year, i.e., X = 0 ∴ P(X = 0) = p(0) = ⁵C₀(0.01)⁰ (0.99)^5-0 = 1 × 1 × (0.99)⁵ = 0.9510 (ii) At least one of them will die within a year, i.e., X ≥ 1 Now, P(X ≥ 1) = p(1) + p (2) + p(3) + p( 4) + p (5) = 1 – p(0) = 1 – 0.9510 = 0.049

If the probability that any 50 year old person will die within a year is 0.01, find the probability that out of a group of 5 such persons(i) none of them will die within a year(ii) at least one of them will die within a year.

Answer»

p = Probability that any 50 year old person will die within a year = 0.01

∴ q = 1 – p = 1 – 0.01 = 0.99
X is binomial random variable.

∴ Putting, n = 5, p = 0.01, q = 0.99 in
P(X = x) = p(x) = ⁿC_xp^xq^n-x
P(X = x) = p(x) = ⁵C_x(0.01)^x(0.99)^n-x

(i) None of the 50 year old person will die within a year, i.e.,
X = 0
∴ P(X = 0) = p(0) = ⁵C₀(0.01)⁰ (0.99)^5-0
= 1 × 1 × (0.99)⁵
= 0.9510

(ii) At least one of them will die within a year, i.e., X ≥ 1
Now, P(X ≥ 1)
= p(1) + p (2) + p(3) + p( 4) + p (5)
= 1 – p(0)
= 1 – 0.9510
= 0.049