Q.81. Probability

Question 81.
A batch of experimental electronic components is such that each component has (independently) only a 56% chance of functioning correctly. Two correctly functioning components are required for a space mission, but they can only be tested in orbit. How many of the components must be taken into space to be at least 98% of having two that function?

Answer 81.
Probability of selected component functioning correctly p = 0.56.
Probability of selected component functioning incorrectly q = 0.44.

If n components are selected and at least 2 of them functioning correctly should have a probability ≥ 0.98, then
required probability = 1 - probability that exactly 0 or 1 are correctly functioning
=> 1 - nC0 (0.56)^0 * (0.44)^n - nC1(0.56)^1*(0.44)^(n - 1) ≥ 0.98
=> nC0 (0.56)^0 * (0.44)^n + nC1(0.56)^1*(0.44)^(n - 1) ≤ 0.02
=> (0.44)^n + (0.56n)*(0.44)^(n - 1) ≤ 0.02
=> (0.44)^(n -1) * [ 0.44 + 0.56n ] ≤ 0.02

Using iterative process,
n = 10
=> (0.44)^9 * [0.44 + (0.56)*10] = 0.00373
n = 9
=> (0.44)^8 * [0.44 + (0.56)*9] = 0.0077
n = 8
=> (0.44)^7 * [0.44 + (0.56)*8] = 0.016
n = 7
=> (0.44)^6 * [0.44 + (0.56)*7] = 0.032
Thus, n = 8 is the answer.

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