Calculating Probability: Sum of 79 Rolls and Exceeding 300

[AKBAR]

I'm confused about this question.

The problem: The sum of the rolls of a fair die exceed 300. Find the probability that at least 80 rolls were necessary.

The solution we were given is that this event is equivalent to the sum of 79 rolls being less than or equal to 300. But I don't get it. The sum of 79 rolls could be 79, assuming you rolled a one every time. Then the 80th roll would not exceed 300. There are plenty of cases of the sum of 79 rolls being less than 300 where the 80th roll would not exceed 300. Can anyone help explain this to me?

Thanks!

 

[wildman]

AT LEAST 80 rolls. That means 80 or MORE. Your example has more than 80 rolls. So it is at least 80 rolls.

 

[tacman]

The solution given is correct, but it may not be immediately apparent why. Let's break down the problem and the solution step by step to help clarify any confusion.

First, the problem states that we are rolling a fair die, which means that each roll has an equal chance of landing on any of the six sides. This also means that the sum of the rolls will follow a normal distribution.

Next, we are asked to find the probability that at least 80 rolls were necessary for the sum to exceed 300. This is equivalent to finding the probability that the sum of 79 rolls is less than or equal to 300.

Why is this the case? Well, think about it this way: if the sum of 79 rolls is less than or equal to 300, then the 80th roll must be greater than 300 for the total sum to exceed 300. This is because each roll has an equal chance of landing on any of the six sides, so it is unlikely that the 80th roll alone will exceed 300. Therefore, we can simplify the problem by focusing on the sum of 79 rolls instead of trying to account for the 80th roll separately.

Now, to address your concern about the sum of 79 rolls potentially being 79 if you rolled a one every time, let's look at the probability distribution for the sum of 79 rolls. The minimum possible sum would be 79 (if you rolled a one every time), and the maximum possible sum would be 474 (if you rolled a six every time). However, the most likely sum would be around 280-300, with a standard deviation of about 17. This means that the probability of the sum being exactly 300 is very small, and the probability of it being less than 300 is even smaller. So, while it is technically possible for the sum of 79 rolls to be less than 300, it is highly unlikely.

I hope this helps clarify the solution and why it makes sense to focus on the sum of 79 rolls instead of trying to account for the 80th roll separately. If you have any further questions or concerns, please don't hesitate to ask. Good luck with your studies!