Math Problem Statement
Solution
The problem asks for the probability that at least one of the 18 students selects a U.S. president who died in office.
To solve this, we will use the complement rule, which is often easier in "at least one" problems.
Step-by-Step Solution:
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Complement Probability: First, we calculate the probability that none of the students select a president who died in office. We subtract this from 1 to get the probability that at least one does.
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Basic Setup:
- There are 43 U.S. presidents, and 8 of them died in office.
- The probability of selecting a president who did not die in office is:
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No Student Picks a Deceased President: If no student picks a president who died in office, all 18 students must pick from the 35 who didn't. The probability that all 18 students avoid picking a deceased president is:
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Complement (At Least One Death): Now, the probability that at least one student picks a president who died in office is:
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Calculation: Let's compute: Then, raise this to the 18th power: Finally, subtract from 1:
Final Answer:
The probability that at least one student selects a president who died in office is approximately 0.96 (rounded to the nearest hundredth).
Would you like more details or have any questions? Here are some follow-up questions:
- How is the complement rule useful in probability problems?
- What happens to the probability if the number of students increases?
- How does the probability change if fewer presidents had died in office?
- Can you explain why we used powers of probability in this solution?
- What if there were restrictions on the number of times a president could be chosen?
Tip: The complement rule simplifies many "at least one" problems by shifting focus to the opposite scenario.
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Math Problem Analysis
Mathematical Concepts
Probability
Complement Rule
Independent Events
Formulas
P(no death) = (35/43)
P(no death for all 18) = (35/43)^18
P(at least one death) = 1 - (35/43)^18
Theorems
Complement Rule
Suitable Grade Level
Grades 9-12
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