In a certain lottery, ten balls are numbered #1 through #10. Two winning balls will be drawn without replacement. Billy plays the lottery and picks the numbers 9 and 10.
What is the probability Billy Bob will have two winning numbers?
I got following two answers from chatGPT. Both the approaches of the chatGPT seems correct to me, but the final answer is different. Since there can be only one answer, However I am not able to reason against any of the approaches. Here are the 2 answers of ChatGpt:
1] Billy Bob has chosen two specific numbers, 9 and 10. There are a total of 10 balls numbered from 1 to 10.
To find the probability that both of his chosen numbers (9 and 10) will be the winning balls, we need to consider the following:
The probability that the first ball he draws is a winning number (9) is 1/10, as there's one favorable outcome out of 10 possible outcomes.
Since the first ball was drawn without replacement, there are now 9 balls left, and the probability that the second ball he draws is a winning number (10) is 1/9.
To find the overall probability, you multiply these probabilities:
(1/10) * (1/9) = 1/90
So, the correct probability that Billy Bob will have two winning numbers is 1/90.
2] To calculate the probability of Billy Bob having two winning numbers, we'll consider the total number of ways to choose two balls out of ten without replacement. We already calculated this as "C(10, 2)" or "10 choose 2," which is 45 ways.
Now, we need to consider the number of favorable outcomes, which is the number of ways Billy Bob can select the two specific numbers he chose (9 and 10), and that is 1 way.
So, the probability is:
Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes) Probability = 1 / 45
I apologize for the previous incorrect response. The correct probability that Billy Bob will have two winning numbers is 1/45.
which one is correct?
The first answer given by ChatGPT obviously is wrong. When the first number is picked up, either $9$ or $10$ is a winning number, so the probability is $\frac{2}{10}$, not $\frac{1}{10}$. Then, the probability that the rest one is picked is $\frac{1}{9}$. So the final answer should be $\frac{2}{10}\times \frac{1}{9}=\frac{1}{45}$.