Question 1
You walk into a barn and see a collection of spiders, chickens, and cows. You notice that there are 520 legs in total. The number of chickens is twice the number of cows and the number of spiders is twice the number of chickens. Compute the number of spiders.

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Question 2

You are on an island inhabited by two types of people:

1. Knights who only tell the truth, and
2. Knaves who only tell lies.

Two people, A and B, stand before you.

• Person A says, "We are both the same type."
• Person B says, "We are each a different type."

What type is each person?

Options:

• A is a knave, B is a knight
• A is a knight, B is a knave
• A is a knight, B is a knight
• A is a knave, B is a knave
• Not enough information

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Question 4
Consider the following table of profits in dollars by year at a university bookstore:

Year	Tech	Books	Café	Clothing	Total
2016	8,039	6,136	5,275	8,698	28,148
2017	6,576	6,475	6,648	7,023	26,722
2018	5,637	9,497	7,654	7,918	30,706
2019	5,393	8,373	9,352	8,402	31,520

We pull a sector’s annual profitability at random and find that they made more than $8,000. Compute the probability that it was from 2019.

Note: Express your answer as a fraction in simplest form. Enter the numerator in the first blank and the denominator in the second blank.

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Question 5
Factory A makes 40% red widgets and 60% black widgets.
Factory B makes 80% red widgets and 20% black widgets.

Two widgets are sampled uniformly at random from one of the companies, also selected uniformly at random. The widgets are both red. Compute the probability they were from Factory A.

Note: Express your answer as a fraction in simplest form. Enter the numerator in the first blank and the denominator in the second blank.

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Question 6
You roll three fair 6-sided dice.

• If they all show the same number, you earn $20.
• If exactly two of the numbers are the same, you earn $10.
• If all of the numbers are different, you lose $2.

Compute your expected return per roll in dollars.

Note:
Express your answer to the nearest cent. Numbers 0-4 round down and 5-9 round up.

Answer format: $__________

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Question 7
Assume that in a bakery, each customer buys only one item at a time. There is a 70% chance a customer will buy a croissant and a 30% chance a customer will buy a muffin.

There are only 2 muffins left, and 5 people are still waiting in line. Compute the probability that these two muffins will be sufficient, i.e., no customer will want a muffin and find that there are none left.

Note:
Express your answer as a percentage to the nearest whole percent. Numbers 0-4 round down and 5-9 round up.

Answer format: ________%

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Question 8
Asta, Bronya, and Clara all have different numbers of cats. They had the following conversation:

1. Bronya to Clara: You have the most cats.
2. Asta to Bronya: I have exactly 30% more cats than you.
3. Asta to Clara: The number of cats you have is the average between the number of cats I have and the number of cats Bronya has.
4. Clara to Asta: You have at least 4 more cats than me.

However, not all of these statements are true. When speaking to someone with fewer cats, the speaker always tells the truth. Otherwise, when speaking to someone with more cats, the speaker always lies.

Question: How many cats does Bronya have?

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Question 9
Three cats are competing in a jumping contest:

• The most athletic cat wins with probability 34\frac{3}{4}43​,
• The least athletic cat wins with probability 116\frac{1}{16}161​, and
• The remaining cat wins with probability 316\frac{3}{16}163​.

You picked a cat uniformly at random to cheer on, but it did not win. Compute the probability that you picked the most athletic cat.

Note:
Express your answer as a fraction in simplest form. Enter the numerator in the first blank and the denominator in the second blank.

Answer format:
__________ / __________

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