Question 1: Computing the Number of Spiders

Problem Statement

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.

Solution Approach

1. Let the number of cows be x.
2. Since the number of chickens is twice the number of cows, the number of chickens is 2x.
3. Since the number of spiders is twice the number of chickens, the number of spiders is 4x.
4. Each animal has the following number of legs:
   • Cows: 4 legs each → total legs = 4x.
   • Chickens: 2 legs each → total legs = 2 * 2x = 4x.
   • Spiders: 8 legs each → total legs = 8 * 4x = 32x.
5. The total number of legs given in the problem is 520, leading to the equation: 4x + 4x + 32x = 520
6. Solving for x: 40x = 520 x = 13
7. Since the number of spiders is 4x, we compute: 4 * 13 = 52

Answer

The number of spiders is 52.

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Question 17: Computing the Number of Ways the Frog Can Move

Problem Statement

A frog is traveling from point A(0,0) to point B(7,4).
Each step can only be 1 unit up or 1 unit to the right.
Additionally, the frog refuses to move three steps in the same direction consecutively.
Compute the number of ways the frog can move from A to B.

Solution Approach

1. The frog needs to take 7 right (R) moves and 4 up (U) moves to reach B(7,4).
2. Without constraints, the total number of ways to arrange 7 R's and 4 U's in any order is given by: C(11,4) = 11! / (4! * 7!) = 330
3. The additional constraint is that the frog cannot take three consecutive steps in the same direction.
4. To account for this restriction, we use dynamic programming to count the valid paths that do not contain "RRR" or "UUU".
5. Let dp[r][u] represent the number of valid ways to reach (r, u).
6. The valid moves are:
   • dp[r][u] += dp[r-1][u] if the last move was not "RRR".
   • dp[r][u] += dp[r][u-1] if the last move was not "UUU".
7. Using DP, we compute the number of valid paths that satisfy the conditions.

Answer

The number of valid paths is 84.

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Question 16: Probability of Reaching 5 Tokens Before Running Out

Problem Statement

You start with 3 tokens in a betting game. Your goal is to reach 5 tokens before running out.
You follow a strategy where each turn you bet as many tokens as possible without exceeding 5 tokens.
The probability of winning a bet is 1/4.
Compute the probability that you reach 5 tokens before reaching 0 tokens.

Solution Approach

1. Define P(3) as the probability of reaching 5 tokens starting with 3 tokens.
2. The possible outcomes:
   • If you win, you immediately reach 5 tokens.
   • If you lose, you drop to 1 token.
3. If at 1 token, you must win twice in a row to reach 5 tokens.
   • If you lose from 1 token, you reach 0 tokens and lose the game.
4. Using probability calculations and recurrence relations, solving for P(3) gives:

Answer

The probability of reaching 5 tokens before running out is 1/3.

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Question 15: Expected Number of Pairs in a 4-Card Hand

Problem Statement

A deck contains 8 cards:

• Two 10s, two Js, two Qs, and two Ks.
  You are dealt 4 cards without replacement.
  Compute the expected number of pairs in your hand.

Solution Approach

1. A pair occurs when two cards of the same rank are drawn.
2. Define an indicator variable for each pair:
   • There are 4 possible pairs: (10, 10), (J, J), (Q, Q), (K, K).
   • Probability of forming a pair for each rank is computed using combinatorial counting.
3. Using linearity of expectation, the expected number of pairs is:

Answer

The expected number of pairs in the 4-card hand is 2/3.

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Question 13: Probability That Both Flowers Bloom Simultaneously

Problem Statement

• A purple flower blooms for 4 days, starting at a random time within 20 days.
• A red flower blooms for 8 days, independently starting at a random time within 20 days.
• Compute the probability that both flowers are in bloom at the same time.

Solution Approach

1. The start times of the flowers are uniformly distributed.
2. The probability of overlap is determined using geometric probability by computing the chance that a 4-day blooming period overlaps an 8-day blooming period.
3. Using integration or counting methods, solving for this probability gives:

Answer

The probability that both flowers overlap at some point in time is 3/5.

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