Question

Suppose that a satellite defense system is established in which four satellites acting independently each have a 0.9 chance of detecting an incoming ballistic missile. a) What is the probability that all 4 satellites will correctly detect the next incoming missile? b) What is the probability that at least one of the four satellites will correctly detect the next incoming missile?

Answer 1

Answer: a) 0.6561

b) 0.9999

Step-by-step explanation:

We use Binomial distribution here (Since each satellite is independent of each other.).

In  Binomial distribution :$P(x)=^nC_xp^x(1-p)^{n-x}$

, where P(X) = probability of getting x successes in n trials.

p= probability of getting success in each trial.

As per given , we have

n= 4

p= 0.9

Let x = Number of  satellites will correctly detect the next incoming missile.

a) The probability that all 4 satellites will correctly detect the next incoming missile:

$P(x)=^4C_4(0.9)^4(1-0.9)^{0}$

$P(x)=(1)(0.9)^4(1)=0.6561$

∴the probability that all 4 satellites will correctly detect the next incoming missile=0.6561

b) The probability that at least one of the four satellites will correctly detect the next incoming missile :

$P(X\geq1)=1-P(X<1)\\\\=1-P(X=0)\\\\=1-^4C_0(0.9)^0(1-0.9)^4\\\\=1-(1)(1)(0.1)^4\\\\=1-0.0001\\\\=0.9999$

∴the probability that at least one of the four satellites will correctly detect the next incoming missile = 0.9999