Leaving Cert Mathematics Counting/Permutations: Peter and Niamh go to a large sc

Peter and Niamh go to a large school - Leaving Cert Mathematics - Question 1 - 2012

Question 1

Peter and Niamh go to a large school. One morning, they arrive early. While they are waiting, they decide to guess whether each of the next three students to come in... show full transcript

Worked Solution & Example Answer:Peter and Niamh go to a large school - Leaving Cert Mathematics - Question 1 - 2012

Step 1

Write out the sample space showing all the possible outcomes.

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Answer

The possible outcomes for the genders of the next three students can be represented as follows:

BBB (Boy, Boy, Boy)
BBG (Boy, Boy, Girl)
BGB (Boy, Girl, Boy)
BGG (Boy, Girl, Girl)
GBB (Girl, Boy, Boy)
GBG (Girl, Boy, Girl)
GGB (Girl, Girl, Boy)
GGG (Girl, Girl, Girl)

Thus, the complete sample space is: BBB, BBG, BGB, BGG, GBB, GBG, GGB, GGG.

Step 2

Peter says these outcomes are equally likely. Niamh says they are not. What do you need to know about the students in the school to decide which of them is correct?

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Answer

To determine whether Peter or Niamh is correct, we need to know the proportion of boys and girls in the school. If the number of boys and girls is approximately equal, then the outcomes may be considered equally likely. However, if one gender is significantly more prevalent than the other, the probabilities of each outcome would differ.

Step 3

If all the outcomes are equally likely, what is the probability that the three students will be two girls followed by a boy?

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Answer

If all outcomes are equally likely, the probability of selecting two girls followed by one boy (GGB) can be calculated as:

$P(GGB) = \frac{1}{8} = 0.125 \text{ or } 12.5\%$

Step 4

Niamh guesses that there will be at least one girl among the next three students. Peter guesses that the next three students will be either three boys or two boys and a girl. Who is more likely to be correct, assuming all outcomes are equally likely? Justify your answer.

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Answer

To assess their guesses:

For Niamh's guess (at least one girl), we can calculate the complementary probability (all boys):
- Probability of all boys (BBB): $P(\text{at least one girl}) = 1 - P(BBB) = 1 - \frac{1}{8} = \frac{7}{8} = 87.5\%$
For Peter's guess (three boys or two boys and one girl):
- Probability for three boys (BBB): $P(BBB) = \frac{1}{8}$
- Probability for two boys and one girl (BBG, BGB, GBB): $P(BBG) + P(BGB) + P(GBB) = \frac{3}{8}$
- Total probability for Peter's guesses: $P(\text{Peter's guess}) = P(BBB) + P(\text{two boys and one girl}) = \frac{1}{8} + \frac{3}{8} = \frac{4}{8} = 50\%$

Since Niamh's guess has a higher probability (87.5%) compared to Peter's 50%, Niamh is more likely to be correct.

Peter and Niamh go to a large school - Leaving Cert Mathematics - Question 1 - 2012

Worked Solution & Example Answer:Peter and Niamh go to a large school - Leaving Cert Mathematics - Question 1 - 2012

Write out the sample space showing all the possible outcomes.

Peter says these outcomes are equally likely. Niamh says they are not. What do you need to know about the students in the school to decide which of them is correct?

If all the outcomes are equally likely, what is the probability that the three students will be two girls followed by a boy?

Niamh guesses that there will be at least one girl among the next three students. Peter guesses that the next three students will be either three boys or two boys and a girl. Who is more likely to be correct, assuming all outcomes are equally likely? Justify your answer.

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