The probability that a bomb dropped from a plane over a bridge will hi

1.	The probability that a bomb dropped from a plane over a bridge will hit the bridge is \(\frac{1}{5}\). Two bombs are enough to destroy the bridge. If 6 bombs are dropped on the bridge, find the probability that the bridge will be destroyed.
Answer» p = Probability that a bomb dropped over a bridge = \(\frac{1}{5}\) = 0.2 X is binomial random variable. ∴ P(x = x) = p(x) = ⁿC_xp^xq^n-x Putting, n = 6, p = \(\frac{1}{5}\) = 0.2, q = \(\frac{4}{5}\) = 0.8 P(X = x) = p(X) = ⁶C_x(0.2)^x(0.8)^6-x Two bombs are enough to destroy the bridge. i.e., X ≥ 2 ∴ P(X ≥ 2) = 1 – p[X ≤ 1] = 1 – [p(0) + p(1)] ………… (1) ∴ p(0) = ⁶C₀(0.2)⁰(0.8)⁶ = 1 × 1 × 0.2621 = 0.2621 ∴ p(1) = ⁶C₁(0.2)¹(0.8)⁵ = 6 × 0.2 × 0.32768 = 0.3932 Putting the values of p(0) and p(1) in the result (1), P[X ≥ 2] = 1 – [0.2621 + 0.3932] = 1 – 0.6553 = 0.3447 Hence, the probability that the bridge will be destroyed obtained is 0.3447.

The probability that a bomb dropped from a plane over a bridge will hit the bridge is \(\frac{1}{5}\). Two bombs are enough to destroy the bridge. If 6 bombs are dropped on the bridge, find the probability that the bridge will be destroyed.

Answer»

p = Probability that a bomb dropped over a bridge = \(\frac{1}{5}\) = 0.2
X is binomial random variable.

∴ P(x = x) = p(x) = ⁿC_xp^xq^n-x

Putting, n = 6, p = \(\frac{1}{5}\) = 0.2, q = \(\frac{4}{5}\) = 0.8

P(X = x) = p(X) = ⁶C_x(0.2)^x(0.8)^6-x

Two bombs are enough to destroy the bridge. i.e., X ≥ 2

∴ P(X ≥ 2) = 1 – p[X ≤ 1]

= 1 – [p(0) + p(1)] ………… (1)

∴ p(0) = ⁶C₀(0.2)⁰(0.8)⁶

= 1 × 1 × 0.2621 = 0.2621

∴ p(1) = ⁶C₁(0.2)¹(0.8)⁵

= 6 × 0.2 × 0.32768

= 0.3932

Putting the values of p(0) and p(1) in the result (1),

P[X ≥ 2] = 1 – [0.2621 + 0.3932]

= 1 – 0.6553

= 0.3447

Hence, the probability that the bridge will be destroyed obtained is 0.3447.