Answer :
Let's carefully analyze each situation and determine if it is a correct example of randomness and equal probability.
### Situation #1
A high school student is deciding whether to first clean his room or do his homework. He decides by picking a tile at random from a bag of lettered tiles. If he picks a consonant, he will clean his room, and if he picks a vowel, he will do his homework.
Analysis: Picking a tile at random from a bag of lettered tiles is a good example of randomness if the vowels and consonants are equally represented in the bag. Assuming there are an equal number of vowels and consonants, both outcomes have an equal probability.
Conclusion: This is a correct situation.
### Situation #2
There are four candidates eligible for a vacancy at a company. Ignoring qualifications and experience, the recruitment manager decides which candidate to hire by writing their names on pieces of paper, shuffling the papers, and drawing one at random.
Analysis: Writing names on pieces of paper, shuffling them, and drawing one is a proper method to ensure randomness, assuming there is no bias in the shuffling process.
Conclusion: This is a correct situation.
### Situation #3
Five cousins are deciding which board game to play. They each write their preferred game on a different section of a spinner that has five equally-sized sections. They spin the spinner and will play the game written on the section where the spinner lands.
Analysis: Using a spinner with equally-sized sections and spinning it to make a decision ensures that each game has an equal probability of being chosen, provided the spinner is unbiased.
Conclusion: This is a correct situation.
### Situation #4
To determine who can choose the spot for a picnic, Kate picks a tile at random from a collection of five tiles, numbered 2 through 6. Kate chooses the spot if she picks a prime number, and Charles chooses if Kate picks a composite number.
Analysis: The tiles are numbered 2 through 6, and the prime numbers in this range are 2, 3, and 5, while the composite numbers are 4 and 6. This means there are three prime numbers and two composite numbers, leading to unequal probabilities.
Conclusion: This is not a correct situation because the probabilities are not equal.
### Situation #5
Five roommates all want to attend an event, but they only have four invites. To decide who will attend, they shuffle a set of five cards consisting of four aces and a king. Each roommate is randomly dealt a card. The ones dealt an ace will attend the event.
Analysis: Shuffling a set of five cards, where four are aces and one is a king, and dealing them randomly ensures an equal probability for each outcome.
Conclusion: This is a correct situation.
### Summary
Combining all the correct situations, we have:
1. Situation #1
2. Situation #2
3. Situation #3
4. Situation #5
So, the correct situations are: [1, 2, 3, 5].
### Situation #1
A high school student is deciding whether to first clean his room or do his homework. He decides by picking a tile at random from a bag of lettered tiles. If he picks a consonant, he will clean his room, and if he picks a vowel, he will do his homework.
Analysis: Picking a tile at random from a bag of lettered tiles is a good example of randomness if the vowels and consonants are equally represented in the bag. Assuming there are an equal number of vowels and consonants, both outcomes have an equal probability.
Conclusion: This is a correct situation.
### Situation #2
There are four candidates eligible for a vacancy at a company. Ignoring qualifications and experience, the recruitment manager decides which candidate to hire by writing their names on pieces of paper, shuffling the papers, and drawing one at random.
Analysis: Writing names on pieces of paper, shuffling them, and drawing one is a proper method to ensure randomness, assuming there is no bias in the shuffling process.
Conclusion: This is a correct situation.
### Situation #3
Five cousins are deciding which board game to play. They each write their preferred game on a different section of a spinner that has five equally-sized sections. They spin the spinner and will play the game written on the section where the spinner lands.
Analysis: Using a spinner with equally-sized sections and spinning it to make a decision ensures that each game has an equal probability of being chosen, provided the spinner is unbiased.
Conclusion: This is a correct situation.
### Situation #4
To determine who can choose the spot for a picnic, Kate picks a tile at random from a collection of five tiles, numbered 2 through 6. Kate chooses the spot if she picks a prime number, and Charles chooses if Kate picks a composite number.
Analysis: The tiles are numbered 2 through 6, and the prime numbers in this range are 2, 3, and 5, while the composite numbers are 4 and 6. This means there are three prime numbers and two composite numbers, leading to unequal probabilities.
Conclusion: This is not a correct situation because the probabilities are not equal.
### Situation #5
Five roommates all want to attend an event, but they only have four invites. To decide who will attend, they shuffle a set of five cards consisting of four aces and a king. Each roommate is randomly dealt a card. The ones dealt an ace will attend the event.
Analysis: Shuffling a set of five cards, where four are aces and one is a king, and dealing them randomly ensures an equal probability for each outcome.
Conclusion: This is a correct situation.
### Summary
Combining all the correct situations, we have:
1. Situation #1
2. Situation #2
3. Situation #3
4. Situation #5
So, the correct situations are: [1, 2, 3, 5].
