1. **Stating the problem:** We have a Bingo game with letters B, I, N, G, O each associated with a range of numbers: - B: 1-15 - I: 16-30 - N: 31-45 - G: 46-60 - O: 61-75 We analyze three probability models: - Viet's model: probability of each number being called first. - Quinn's model: probability of each letter being called first. - Lucy's model: probability the first number drawn is even or odd. 2. **Part A: Comparing Viet's and Quinn's probability models** - Viet's model assumes each number from 1 to 75 is equally likely. - Probability of any specific number being called first is $$\frac{1}{75}$$. - Quinn's model assumes each letter (B, I, N, G, O) is equally likely. - Probability of any letter being called first is $$\frac{1}{5}$$. 3. **Part B: Lucy's model for even or odd first number** - Total numbers: 75 - Count of even numbers: numbers divisible by 2 from 1 to 75. - Even numbers: 2,4,6,...,74. Number of even numbers is $$\frac{74}{2} = 37$$. - Odd numbers: 75 - 37 = 38. - Probability first number is even: $$\frac{37}{75}$$. - Probability first number is odd: $$\frac{38}{75}$$. 4. **Part C: Changing the game to 90 numbers** - New ranges for letters (assuming equal distribution): - B: 1-18 - I: 19-36 - N: 37-54 - G: 55-72 - O: 73-90 - Viet's model: probability of each number being called first is now $$\frac{1}{90}$$. - Quinn's model: probability of each letter being called first remains $$\frac{1}{5}$$ because letters count is unchanged. - Lucy's model: - Even numbers from 1 to 90: $$\frac{90}{2} = 45$$. - Odd numbers: 90 - 45 = 45. - Probability first number even: $$\frac{45}{90} = \frac{1}{2}$$. - Probability first number odd: $$\frac{45}{90} = \frac{1}{2}$$. **Summary:** - Viet's model probability changes inversely with total numbers. - Quinn's model probability stays constant as letter count is fixed. - Lucy's model probabilities adjust based on count of even/odd numbers in total.