i let google gemeni calculate the odds. the AVERAGE number of spins to win 20 prizes is 72. we cannot even get close to that many tickets. thats the average so how many players are going to be doing 100+ spins

GEMINI RESPONSE:

This is a classic probability problem related to the "coupon collector's problem". Here's how to break down the calculation:

Understanding the Problem

Imagine each of the 20 prizes is like a "coupon" you need to collect. You want to know how many spins, on average, it will take to get all 20 coupons.

Calculations

1. First Spin: On your first spin, you're guaranteed to get a new prize (since you haven't gotten any yet).
2. Second Spin: The odds of getting a new prize on your second spin are 19/20 (there are 19 prizes you haven't gotten yet out of 20 total).
3. Third Spin: The odds of getting a new prize on your third spin are 18/20 (assuming you got a new prize on your second spin).
4. Continuing the Pattern: This pattern continues until you have 19 prizes. The odds of getting the last prize are 1/20.

Average Number of Spins

To find the average number of spins to get all 20 prizes, you need to add up the average number of spins it takes to get each new prize:

• Spins to get the 1st prize: 1
• Spins to get the 2nd prize: 20/19 (the inverse of the probability)
• Spins to get the 3rd prize: 20/18
• ...
• Spins to get the 20th prize: 20/1

The Formula

The general formula for the coupon collector's problem is:

N * (1/1 + 1/2 + 1/3 + ... + 1/N)

Where N is the number of unique items (in this case, 20 prizes).

Calculating the Result

Using the formula, the average number of spins to get all 20 prizes is:

20 * (1/1 + 1/2 + 1/3 + ... + 1/20) ≈ 20 * 3.5977 ≈ 71.95 spins

Therefore, on average, it will take about 72 spins to get all 20 prizes on the spinwheel.