Question

7. Records show that a treatment for a rare disease is effective 70% of the time. Suppose that a random sample of 18 patients undergoes this treatment. (a) Find the probability that the treatment will be successful for i) exactly 13 patients (ii) under 10 patients (ii) at least 13 patients iv) 10 to 15 patients inclusive b) Find: ) the mean (expected) number patients for which the treatment will be successful (ii) the variance and standard deviation.

Answer 1

here this is binomial distribution with parameter n=18 and p=0.7

a) i)P(X=13)=¹⁸C₁₃(0.7)¹³(0.3)⁵ =0.2017

ii) P(X<10)=$\sum_{x=0}^{9}\binom{18}{x}(0.7)^{x}(0.3)^{18-x}$ =0.0596

iii) P(X>=13)=$\sum_{x=13}^{9}\binom{18}{x}(0.7)^{x}(0.3)^{18-x}$=0.5344

iv) P(10<=X<=15)=$\sum_{x=10}^{15}\binom{18}{x}(0.7)^{x}(0.3)^{18-x}$ =0.8805

b) i) mean =np=18*0.7=12.6

ii) variance =np(1-p)=3.78

std deviation =sqrt(3.78)=1.944