Probabilities and equally-possible scenarios

[mike.fleischmann]

I am confused about how to understand and put together two conceptually similar, yet distinct possibilities that have equally-probable outcomes. Consider the following Check Your Skills problem from page 134 of guide 5 ("Word Problems"): If a die is rolled twice, what is the probability that it will land on an even number at least once?

I know that the way to answer this is to 1) calculate the probability of never landing on an even number through two rolls and then 2) subtract it from 1:

1/2 x 1/2 = 1/4 chance of never landing on an even number
1 - 1/4 = 3/4 chance of landing on an even number at least once

However, though I know how to simply answer the question, I think I don't fully understand some of the underlying concepts.

By similar math, there is a 3/4 chance of landing an odd number at least once. Thus, the probabilities of at least one even number or at least one odd number are the same. So, I understand how the math leads to this outcome -- but what is the best way to understand how these probabilities co-exist? Here's where I get confused:

- The probability of landing a die on an even or odd number at least once, through any amount of rolls, has to be one 1.
- However, the two events described above are independent events. Thus - to determine the probability that landing on an even number at least once and landing on an odd number at least once will both occur - I multiply the two probabilities together: 3/4 x 3/4 = 9/16.
- Additionally, these two outcomes may be understood as non-mutually exclusive "or" probabilities -- i.e., at least one even or at least one odd - meaning that their shared probability may be calculated as P(even) + P(odd) - P(even and odd): 3/4 + 3/4 -9/16 = 15/16

This all is a mess, and clearly I'm missing something. I'm wondering if anyone can clear this up for me. Thanks.

[tommywallach]

These are not independent. Once you say "at least once" you are including both the possibility of ONCE and TWICE. If that's the case, you are no longer describing two independent things when you say "at least once odd" and "at least once even." You're double counting possibilities.

Hope that helps! If not, let me know and I can keep trying.

Also, I would add, you're basically connecting up piece after piece when you aren't trying to answer a question. I would fixate instead on individual questions that confuse you, as opposed to trying to throw a bunch of words together yourself and then wondering why they don't yet make sense.

-t