A fast-food restaurant is running a scratch-and-win contest. Customers scratch six squares on a card and try to match the six numbers, randomly selected from 1 to 20, that are printed across the top of the card.
• First prize is awarded if all six numbers match.
• Second prize is awarded if five of the six numbers match.
• Third prize is awarded if four of the six numbers match.
What are the odds against winning a prize?
Well, let's calculate the odds against winning a prize at this fast-food restaurant scratch-and-win contest.
For the first prize, you would have a 1 in 20 chance for each number, so the odds against winning it would be 20 multiplied by itself six times, which is a whopping 6,400,000 to 1.
For the second prize, you would have to match five out of the six numbers. The first number you choose has a 1 in 20 chance, and the remaining five numbers have a 1 in 19 chance. So the odds against winning the second prize would be 20 multiplied by 19 to the power of five, which results in approximately 3,190 to 1.
For the third prize, you would have to match four out of the six numbers. The first number you choose has a 1 in 20 chance, and the remaining four numbers have a 1 in 19 chance. So the odds against winning the third prize would be 20 multiplied by 19 to the power of four, which gives you roughly 800 to 1.
So the odds against winning any prize at this fast-food restaurant's scratch-and-win contest are definitely not in your favor. But hey, at least you can always rely on the humor in clown bot's responses!
To calculate the odds against winning a prize, we need to calculate the probability of not winning each prize and then subtract it from 1.
First Prize:
The probability of winning the first prize is 1 out of (20 choose 6). So the probability of not winning the first prize is (20-6) choose 6, which is (14 choose 6).
Second Prize:
The probability of winning the second prize is (6 choose 5) multiplied by (14 choose 1), as there are 6 ways to choose 5 matching numbers and 14 ways to choose the non-matching number. So the probability of not winning the second prize is 1 - ((6 choose 5) * (14 choose 1)).
Third Prize:
The probability of winning the third prize is (6 choose 4) multiplied by (14 choose 2), as there are 6 ways to choose 4 matching numbers and 14 choose 2 ways to choose the 2 non-matching numbers. So the probability of not winning the third prize is 1 - ((6 choose 4) * (14 choose 2)).
Now, we can calculate the odds against winning a prize:
Odds against winning a prize = (probability of not winning first prize) * (probability of not winning second prize) * (probability of not winning third prize)
= (14 choose 6) * (1 - ((6 choose 5) * (14 choose 1))) * (1 - ((6 choose 4) * (14 choose 2)))
To get the final probability, subtract the odds against winning a prize from 1:
1 - Odds against winning a prize
Please note that calculating the exact probability value would require precise calculations involving factorials and binomial coefficients, which may not be possible to compute accurately without using a computer program or calculator.
To calculate the odds against winning a prize, we need to know the total number of possible outcomes and the number of successful outcomes (i.e., the number of ways to win a prize).
1. First, let's determine the total number of possible outcomes:
Since each number can range from 1 to 20, there are 20 choices for the first square, 20 choices for the second square, and so on. Therefore, the total number of possible outcomes is 20 x 20 x 20 x 20 x 20 x 20 = 20^6 = 64,000,000.
2. Next, let's determine the number of successful outcomes for each prize category:
A. First prize: For all six numbers to match, there is only one successful outcome.
B. Second prize: To have five of the six numbers match, we need to select one number to not match. There are six ways to choose which number won't match (since there are six numbers in total), and for each of those choices, there are 19 choices for the unmatched number (since it can be any number from 1 to 20, except for the one already chosen). Therefore, there are 6 x 19 = 114 successful outcomes for the second prize.
C. Third prize: To have four of the six numbers match, we need to choose two numbers to not match. There are six ways to choose which two numbers won't match, and for each choice, there are 19 choices for the first unmatched number and 18 choices for the second unmatched number. Therefore, there are 6 x 19 x 18 = 2,052 successful outcomes for the third prize.
3. Finally, let's calculate the odds against winning a prize:
The odds against winning a prize are given by the ratio of unsuccessful outcomes to successful outcomes.
Unsuccessful outcomes = Total number of possible outcomes − Number of successful outcomes.
Odds against winning a prize = Unsuccessful outcomes / Successful outcomes.
Unsuccessful outcomes = 64,000,000 - (1 + 114 + 2,052) = 63,998,833.
Successful outcomes = 1 + 114 + 2,052 = 2,167.
Therefore, the odds against winning a prize are approximately 63,998,833 / 2,167 ≈ 29,532. This means that for every 29,532 tickets played, on average, you can expect to win a prize.