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The track team gives awards for first, second, and third place runners. There are 10 students from school A and 12 students from school B competing.

Which expression represents the probability that all three awards will go to a student from school B?

A. [tex]\(\frac{12^P}{22^P}\)[/tex]
B. [tex]\(\frac{{ }_{12} C_3}{{ }_{22} C_3}\)[/tex]
C. [tex]\(\frac{22 P_3}{22 P_{12}}\)[/tex]
D. [tex]\(\frac{{ }_{22} C _3}{{ }_{22} C _{12}}\)[/tex]

Sagot :

GinnyAnswer
To determine the probability that all three awards will go to students from school B, we need to consider the probabilities of selecting students from school B in three successive selections without replacement.

Here are the steps for calculating this probability:

1. Total Students: The total number of competing students is 10 from school A plus 12 from school B, which sums up to 22 students in total.

2. First Selection: The probability that the first student selected is from school B is [tex]\( \frac{12}{22} \)[/tex].

3. Second Selection: After one student from school B has been chosen, 11 students from school B remain out of the remaining 21 students. So, the probability that the second student selected is also from school B is [tex]\( \frac{11}{21} \)[/tex].

4. Third Selection: After two students from school B have been chosen, 10 students from school B remain out of the remaining 20 students. Therefore, the probability that the third student selected is from school B is [tex]\( \frac{10}{20} \)[/tex] or [tex]\( \frac{1}{2} \)[/tex].

To find the overall probability that all three selections are from school B, we multiply these individual probabilities together:
[tex]\[ \left( \frac{12}{22} \right) \times \left( \frac{11}{21} \right) \times \left( \frac{10}{20} \right) \][/tex]

Now, let's analyze the given options:

1. [tex]\(\frac{12^P}{22^P}\)[/tex] - This option is not correct as it does not represent the probability of selecting students from school B specifically.
2. [tex]\(\frac{{ }_{12} C_3}{{ }_{22} C_3}\)[/tex] - This represents the ratio of the number of ways to choose 3 students out of 12 to the number of ways to choose 3 students out of 22. While related to combinations, this is not the specific sequential probability calculation.
3. [tex]\(\frac{22 P_3}{22 P_{12}}\)[/tex] - This expression is also not correct for the scenario of choosing three students consecutively.
4. [tex]\(\frac{{ }_{22} C _3}{{ }_{22} C _{12}}\)[/tex] - This is an incorrect interpretation for the required probability.

The correct answer is the approach detailed in the steps above, involving the probabilities of successive selections:

[tex]\[ \left( \frac{12}{22} \right) \times \left( \frac{11}{21} \right) \times \left( \frac{10}{20} \right) \][/tex]

This multiplication gives us the probability [tex]\(0.14285714285714285\)[/tex], which matches our objective calculation.
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