Question about the Rule of Product

I’m currently working my way through the statistical part of the Data Analyst path and I got a question about the Mission “Permutations and combinations”, on this screen.

It’s a very simple question, so I think I’m not seeing something small, but here it is:

The question command asks us to find the probability of not getting a (5,5) in two dice rolls.

I thought of two ways to calculate it:

  1. Assuming that the probability of getting a (5,5) is 1 - the probability of getting a (5,5). So, P(not 5,5) = 1 - (1/6)**2 (which I learned from the combining formulas task in the “Solving Complex Probability Problems” mission).

  2. Squaring the chance of not getting a 5 in either roll, so: p(not 5,5) = (5/6)*(5/6) (which emcompasses the multiplication rule from the “Solving Complex Probability Problems”.

  3. Thinking of a (5,5) as in result in 36 possible ones. So it would be P(not 5,5) = 1 - (1/36) - which is the addition rule from the mission “Probability rules”.

However, only answers 1 and 3 work - no surprise, since (1/6)**2 is (1/36), but what’s wrong with the second one ? What’s wrong with the logic I used.

Thanks in advance.

You’re discarding the possibility of getting a 5 in exactly one of the rolls. What you found, as you yourself said, was the probability of not getting a 5 in either roll.

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