Probability of Toy Car in nth Box Given 9/n-1 Boxes Have Toy Cars

In summary, the probability of finding a toy car in the 10th box and any other box in a set of 10 is dependent on the total number of boxes in the universe and the number of boxes that contain toy cars. Without this information, it is not possible to accurately determine the probability. One suggested method is to use the sample data to estimate the probability and calculate a confidence interval.
  • #1
Cemre
14
0
Let's say we have 10 boxes and I open each of them one by one...

I open the 1st box, there is a toy car in it.
I open the 2nd box, there is also a toy car in it.
I open the 3rd box, there is also a toy car in it.
...
I open the 9th box, there is also a toy car in it. :) wow, I got 9 toy cars in 9 boxes...

what is the probability that 10th box also has a toy car in it?

also generalize 10 to any number...
what is the probability that nth box also has a toy car in it, if all n-1 boxes each have a toy car in them.
 
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  • #2
There are lots of ways to do this binomial confidence interval problem. One common way is prob = (# successes + 1) / (# trials + 2), which would suggest a 91% chance.
 
  • #3
Cemre said:
what is the probability that 10th box also has a toy car in it?

Depends on where you're getting the boxes.
 
  • #4
Yes this is a silly question, unless you give some more information there is no well defined answer.

About the best answer I could give (without any additional information) would be,

[tex]P = \frac{m-9}{n-9}[/tex]

Where n is the number of "boxes" in the universe and m<n is number of boxes in the universe that contain toy cars. I know that's not a very useful answer, but you know that if you want a useful answer you have to ask a sensible question right.
 
  • #5
What CRGreathouse is suggesting is to use the sample data to estimate the probability that a single box contains a car. Of course, if you have gotten a car in every box so far, the "maximum likelihood" estimate of that probability is 1 but you can use the sample size to put bounds on a confidence interval for it.
 

1. What is the purpose of studying probability in relation to toy cars in boxes?

The purpose of studying probability in this context is to determine the likelihood of a toy car being in the nth box given that a certain number of boxes before it already have toy cars. This can help us understand the distribution of toy cars in a set of boxes and make predictions about the chances of finding a toy car in a specific box.

2. How is the probability calculated in this scenario?

The probability in this scenario is calculated using the formula P(n) = (9/n-1) * 1/n, where n represents the number of boxes and 9/n-1 represents the number of boxes before the nth box that already have toy cars. This formula assumes that the toy cars are randomly distributed among the boxes.

3. Can the probability be greater than 1?

No, the probability cannot be greater than 1. This would indicate that there is a greater than 100% chance of finding a toy car in the nth box, which is not possible. The maximum probability in this scenario is 1, indicating that there is a 100% chance of finding a toy car in the nth box.

4. How does the number of boxes before the nth box affect the probability?

The number of boxes before the nth box has a direct impact on the probability. As the number of boxes before the nth box increases, the probability of finding a toy car in the nth box decreases. This is because there are fewer toy cars left to distribute among the remaining boxes, making it less likely for the nth box to have a toy car.

5. Are there any limitations to this probability calculation?

Yes, there are some limitations to this probability calculation. It assumes that the toy cars are randomly distributed among the boxes, which may not always be the case in real life. Additionally, it does not take into account any factors that may affect the distribution of toy cars, such as the size or shape of the boxes.

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