Respuesta :
Option A
ANSWER:
The probability that the surgery is successful for exactly 6 patients is 0.088
SOLUTION:
A surgical technique is performed on 10 patients. You are told there is an 80% chance of success. We have to find the probability that the surgery is successful for exactly 6 patients.
For binomial distribution, the probability mass function for random variable X is given as,
[tex]P(X=x)=n C x \times p^{x} \times(1-p)^{n-x}[/tex]
Given, that the total number of patients is n=10
The chance of success for surgical technique is p=80%.
[tex]\text { Probability of getting success p(success) }=\frac{\text {no of favourable outcomes}}{\text {total possible outcomes}}[/tex]
[tex]P(success) = \frac{50}{100} = 0.8[/tex]
x = 6
The probability for surgery is successful for exactly 6 patients is given by
[tex]\mathrm{P}(\mathrm{X}=6) = ^{10} \mathrm{C}_{6} \times(0.8)^{6} \times(1-0.8)^{10-6}[/tex]
[tex]=^{10} \mathrm{C}_{6} \times(0.8)^{6} \times(0.2)^{4}[/tex]
[tex]=\frac{10 !}{(10-6) ! 6 !} \times(0.2621) \times(0.0016)[/tex]
[tex]=\frac{10 \times 9 \times 8 \times 7 \times 6 !}{4 ! \times 6 !} \times 0.0004[/tex]
Solving the factorial we get,
[tex]=\frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} \times(0.0004)[/tex]
[tex]=\frac{5040}{24} \times 0.0004[/tex]
[tex]=210 \times 0.0004[/tex]
P(X=6) = 0.084
Answer is near to option A. Hence probability that the surgery is successful for exactly 6 patients 0.088.
