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.
Input/Output Example
Total legs: 520
Let number of cows = x
Then number of chickens = 2x
Then number of spiders = 2 * (2x) = 4x
Cows have 4 legs: 4 * x
Chickens have 2 legs: 2 * 2x = 4x
Spiders have 8 legs: 8 * 4x = 32x
Total legs = 4x + 4x + 32x = 40x = 520
=> x = 13
Number of spiders = 4x = 52
Question 2
You are on an island inhabited by two types of people:
- Knights who only tell the truth
- 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?
Input/Output Example
Assume A is a knight ⇒ A tells the truth ⇒ B is also a knight
But then B's statement "We are each a different type" is false ⇒ contradiction
Assume A is a knave ⇒ A lies ⇒ "We are both the same" is false ⇒ they are different
Then B must be a knight (telling the truth that they are different)
So A is a knave, B is a knight
Question 3
1000 people were asked about their preferred method of exercise. The following table shows the results, grouped by age.
| 18-22 | 23-27 | 28-32 | 33-37 | Total | |
|---|---|---|---|---|---|
| Run | 54 | 40 | 42 | 66 | 202 |
| Bike | 77 | 68 | 90 | 70 | 305 |
| Swim | 28 | 43 | 50 | 52 | 173 |
| Other | 90 | 78 | 71 | 81 | 320 |
| Total | 249 | 229 | 253 | 269 | 1000 |
You meet a 33-year-old who took the survey. Compute the probability she prefers swimming.
Input/Output Example
Age group: 33-37
Total people in group: 269
People who prefer swimming in group: 52
Probability = 52 / 269
Question 4
Compute the weight of the green triangle in pounds if the following system is balanced and weighs 96 pounds in total.
Input/Output Example
Observe the balanced tree structure:
Break down into parts, work from bottom up.
Let green triangle = x
From balance and symmetry, deduce equations and solve:
Total weight = 96
Each subtree weight can be calculated by summing all elements
After solving through the levels using substitution:
Green triangle = 6
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.
Input/Output Example
P(A) = 0.5
P(B) = 0.5
P(both red | A) = 0.4 * 0.4 = 0.16
P(both red | B) = 0.8 * 0.8 = 0.64
P(both red) = 0.5 * 0.16 + 0.5 * 0.64 = 0.08 + 0.32 = 0.40
P(A | both red) = (0.5 * 0.16) / 0.40 = 0.08 / 0.40 = 1/5
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