Answer:
The mean number of successful surgeries is 1.57.
The variance of the number of successful surgeries is 0.3111.
Step-by-step explanation:
STEP 1
If the tear on the left knee has a rim width of less than 3mm, the probability that the surgery on the left knee will be successful (ls) is 0.90.
That isP(Is)=0.90
.
The probability that the surgery on the left knee will fail (lf) is 0.10. That isP(V)=0.10
.
If the tear on the right knee has a rim width of 3-6 mm, the probability that the surgery on the right knee will be successful (rs) is 0.67. That isP(rs)=0.67
.
The probability that the surgery on the right knee will fail (rf) is 0.33.
That isP(vf)= 0.33
.
Let the random variable X denote the number of successful surgeries. The range of X is{0,1,2)
.
Now find the probabilities associated with the possible values of X. The number of successful surgeries is equal to 0 if the surgeries on both knees fail. Since the surgeries are independent, we have:
P(X = 0)=P(lf and rf)
= P()P(rf)
=(0.10)(0.33)
= 0.033
The number of successful surgeries is equal to 1 if the surgery on one knee is successful and the surgery on the other knee fails, that is
P(X =1)= P((ls and rf) or (if and rs))
= P(Is)P(rf)+P(V)P(rs)
= (0.90)(0.33)+(0.10)(0.67)
= 0.364
The number of successful surgeries is equal to 2 if the surgeries on both knees are successful, that is
P(X = 2)=P(ls and rs)
= P(Is) P(rs)
=(0.90)(0.67)
= 0.603
So, the probability mass function of X is the following
X 0 1 2
f(x) 0.033 0.364 0.603
The mean number of successful surgeries is,
E(X)=∑xP(x)
=(0×0.033) +(1×0.364)+(2×0.603)
=1.57
the mean number of successful surgeries is 1.57
The expected value is obtained by taking the summation of the product of each possible value of the random variable X with its corresponding probabilities. Thus, the mean number of successful surgeries is 1.57.
STEP 2
The variance of the number of successful surgeries is,
Var(X)=∑x²P(x)-(E(x²)
=(0×0.033) + (1 × 0.364) +(2 × 0.603) - (1.57)²
= 2.776 -(1.57)²
=0.3111
The variance of the number of successful surgeries is 0.3111.
