Probability

This problem can be modeled using the binomial probability distribution.
Step 1: Identify the parameters of the binomial distribution.
Number of trials (surgeries), .
Probability of success (recovery), .
Probability of failure (no recovery), .
Number of successful recoveries required, .
Step 2: Apply the binomial probability formula.
The probability of exactly successes in independent Bernoulli trials is given by: We want to find the probability of exactly 2 recoveries ( ): Step 3: Calculate the binomial coefficient.
Step 4: Calculate the probability.
Thus, the probability that exactly two of them will recover is 0.432.

Step 1: Understanding the region .
The inequality represents a circle of radius 2 centered at the origin, but restricted to the first quadrant (i.e., ).
Step 2: Find total area of region .
The total area of a circle is , so the first quadrant portion is:
Step 3: Consider the line within region .
The line divides the circular region into two symmetric parts within the first quadrant: one with , and the other with . So the region where (i.e., ) occupies exactly half of region .
Step 4: Compute the required probability.

We are asked to find the probability that a certain married couple will either serve together or not at all on a committee of 5 chosen from a group of 9 people.
Step 1: Total number of ways to select the committee.
We need to select 5 people from a group of 9. The total number of ways to do this is given by the combination formula , where is the total number of people and is the number of people to be chosen: Step 2: Case 1 - The couple serves together.
If the couple serves together, we treat them as a single unit, so we are left with selecting 3 additional members from the remaining 7 people. The number of ways to do this is: Step 3: Case 2 - The couple does not serve.
If the couple does not serve, we need to select all 5 members from the remaining 7 people. The number of ways to do this is: Step 4: Total favorable outcomes.
The total number of favorable outcomes is the sum of the favorable outcomes from both cases: Step 5: Probability.
The probability is the ratio of favorable outcomes to total outcomes: Thus, the probability that the married couple will either serve together or not at all is .