Answer:
0.028 ; 0.649 ; 4
Step-by-step explanation:
P(correctly activation), p = 0.7
Number of sprinklers, n = 10
q = 1 - p = 1 - 0.7 = 0.3
Using the binomial probability relation :
P(x =x) = nCx * p^x * (1 - p)^(n - x)
Probability that all sprinklers activate correctly;
P(x = 10) = 10C10 * 0.7^10 * 0.3^0
P(x = 10) = 1 * 0.0282475249 * 1
P(x = 10) = 0.028
Probability that atleast 7 will operate correctly :
P(x ≥ 7) = p(x = 7) + p(x = 8) + p(x = 9) + p(x = 10)
P(x ≥ 7) = 0.267 + 0.233 + 0.121 + 0.028
P(x ≥ 7) = 0.649
3.)
Probability of atleast 1 operates :
P(x ≥ 1) = 0.98
1 - probability of 0 operates
1 - p(x =0)
P(x = 0) = nC0 * 0.7^0 * 0.3^n = 0.02
Recall :
nC0 = 1 ;
1 - p(x = 0)
P(x = 0) = 1 * 1 * 0.3^n = 0.02
0.3^n = 0.02 - - - (1)
0.3^3.24 = 0.02 - - - (2)
Comparing (1) and (2)
n = 3.24
Since, n cannot be a fraction ;
Then n is rounded to the next whole number = 4
