Math Problem Statement

As a reward for a record year, a software company is randomly selecting 4 people from its 600 employees for a free trip to Hawaii, but it will not pay for a traveling companion. If John and Jill are married and both are employees, what is the probability that they will both win? (Round your answer to six decimal places.)

Solution

To calculate the probability that John and Jill both win the trip to Hawaii, we can follow these steps:

Step 1: Total ways to choose 4 employees

The company is selecting 4 employees out of 600. The total number of ways to do this is given by the combination formula:

(nr)=n!r!(nr)!\binom{n}{r} = \frac{n!}{r!(n-r)!}

For n=600n = 600 and r=4r = 4: (6004)=6005995985974321=34686151275\binom{600}{4} = \frac{600 \cdot 599 \cdot 598 \cdot 597}{4 \cdot 3 \cdot 2 \cdot 1} = 34686151275

Step 2: Ways for John and Jill to both win

If John and Jill are selected, that leaves 2 more spots to be filled from the remaining 598 employees. The number of ways to choose 2 more people from 598 is:

(5982)=59859721=178203\binom{598}{2} = \frac{598 \cdot 597}{2 \cdot 1} = 178203

Step 3: Probability calculation

The probability that John and Jill both win is the ratio of the favorable outcomes to the total outcomes:

P(John and Jill both win)=(5982)(6004)P(\text{John and Jill both win}) = \frac{\binom{598}{2}}{\binom{600}{4}}

Substitute the values: P(John and Jill both win)=17820334686151275P(\text{John and Jill both win}) = \frac{178203}{34686151275}

Perform the division: P(John and Jill both win)0.000005138P(\text{John and Jill both win}) \approx 0.000005138

Final Answer:

The probability that John and Jill both win is approximately:

0.000005\boxed{0.000005}

Would you like me to verify steps or explain any part in greater detail?


Expanded Questions:

  1. How does the probability change if the total number of employees increases?
  2. What if the company selects 5 employees instead of 4—how does this affect the probability?
  3. How would the probability change if John and Jill were not married (independent outcomes)?
  4. Can we generalize the method to find probabilities for any pair of employees?
  5. What is the complementary probability that at least one of them is not selected?

Tip:

When calculating probabilities, double-check your combinations and ensure all scenarios align with the problem constraints.

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Math Problem Analysis

Mathematical Concepts

Combinatorics
Probability

Formulas

Combination formula: C(n, r) = n! / [r!(n - r)!]
Probability formula: P(Event) = Favorable outcomes / Total outcomes

Theorems

Basic Probability Theorem

Suitable Grade Level

Grades 11-12