Confusion in Probability

I have a confusion regarding probability.

The given problem is:
" Find the probability of getting at least one double-six in 24 throws of two six-sided dice (the two dice are thrown simultaneously). Assign your answer to p_one_double_6 . The table below shows all the outcomes of throwing two six-sided dice."

I solved it this way:
p_one_double_6=(1/36)**24

But correct solution according to the dataquest is
p_one_double_6=1-(35/36)**24

I don’t understand why my solution is considered incorrect?

The link to the page is given below:
https://app.dataquest.io/m/379/solving-complex-probability-problems/7/combining-formulas

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Hi @waqaskhan93

this is the answer for getting one and only one double_six in 24 throws
to answer the question you can compute the probability of the negation of getting at least one double-six , the probability of not getting any double-six in 24 throws is (35/36)**24,
then the probability of getting at least one double-six in 24 throws is 1-(35/36)**24

I still don’t get it. Why would this (1/36)**24 be the probability of getting only one double-six in 24 throws, when I am also multiplying it 24 times with itself?

Hi @waqaskhan93
When two dice are thrown simultaneously, the number of all possible outcomes {(1,1), (1,2), …,(6,6)} can be 6*6 = 36 because each die has 1 to 6 number on its face, the probability of the outcomes (6,6) is given by 1/36 because all outcomes have the same probability of appearing.
To get to the probability of getting only one (6,6) in 24 throws, you would use the multiplication rule of probability

Well, I am using multiplication rule. I am not asking whether to use multiplication rule or not. What I am asking here is that why Dataquest is not considering “(1/36)**24” the correct solution. Dataquest is asking to solve it this way:
“1-(35/36)**24)”. That is what I am concerned about.

(1/36)**24 is actually the probability that you will roll double-six 24 times in a row. Here’s why.

Each of the 24 trials of rolling 2 dice simultaneously is an independent event, so that one roll has no influence on another roll. To find the probability of independent events, we multiply the probabilities of each individual event. Therefore, (1/36)**24 is calculating the probability that we get double-six 24 times.

If we wanted to know the probability of never getting a double-six in all the 24 rolls, we would use (35/36)**24, since there is a 35/36 chance on each roll that we will NOT get a double-six.

Calculating the probability that we get exactly one roll of double-six out of the 24 is more complicated. The probability of it happening on the first roll would be (1/36)*((35/36)**23), but the single occurrence of double-six could really happen on any of the rolls 1-24. It’s even more confusing using the multiplication rule when you consider that there could be more than one double-six among the 24 rolls.

The wording of the problem is asking for the probability that “at least one” (one or more) of the 24 rolls is a double-six. The easiest way to calculate this is to find the opposite – the probability that less than 1 (zero!) of the rolls is a double-six. This is calculated with the (35/36)**24 expression. Then to find the probability of at least one, we subtract the complement from 1. This is where we get the final formula in the solution.

I hope that helps to clear up some of the confusion.

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These two did it for me. I was hoping you would have answered this one, lol.