I have always taught that, in case of two independent events P(1) and P(2), probability of P(1) AND P(2) = P(1) x P(2) and that probability of P(1) OR P(2) is P(1) + P(2). But consider this scenario: Bag A contains 1 red and 7 blue balls.
Bag B contains 1 black and 7 green balls. Select a ball from each bag. What is probability of choosing a blue ball from Bag A OR a green ball from Bag B? Probability of choosing blue from Bag A = 7/8. Probability of choosing green from Bag B = 7/8. So probability of choosing a blue ball from Bag A OR a green ball from Bag B = 7/8 + 7/8 = 14/8 which is more than 1 and clearly incorrect. Where am I going wrong, please?

  • $\begingroup$ Thanks for reply. $\endgroup$ – Keith Smith Mar 27 '17 at 12:58
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    $\begingroup$ While it's great that you realized that something was wrong with your math and came to ask. Please seriously consider learning a lot more about probability before you teach it to anyone. Especially impressionable high-schoolers. $\endgroup$ – DRF Mar 28 '17 at 9:27
  • $\begingroup$ Thanks for your comment. In fact, all of my students have achieved A or A*, so I don't think I'm doing too badly. I posed my problem to a couple of full-time maths teachers, and they couldn't provide the answer. The penny eventually dropped with me, and the answer was straight forward. I note you didn't provide one. $\endgroup$ – Keith Smith Mar 29 '17 at 11:34
  • $\begingroup$ I realize the comment was patronizing but having been the person teaching at the university level who kept being told "But I had straight A's in math in high school. How come I'm struggling so much now?" The fact that neither you and worse yet nor any of your colleagues were able to realize that either you weren't working over the right probability space or you were adding non-mutually exclusive events (depending on how you want to look at it) explains a lot.Sorry but if you are teaching something you should at least know the material. $\endgroup$ – DRF Mar 29 '17 at 12:48
  • $\begingroup$ As a short aside your question is rife with notational issues that take students down completely wrong paths. I now see why a student was so surprised when I docked points for him writing P(0.18+0.4). P(1) OR P(2) makes no sense. Even ignoring the fact that neither 1 nor 2 is a subset of your probability space and so you can't take their probability you have the issue that $P(A)$ ($A$ being an event in your probability space) is a number so when you're saying $P(A)$ or $P(B)$ you're essentially saying $7/8$ or $7/8$ which makes not sense. $\endgroup$ – DRF Mar 29 '17 at 13:06

$P(A\text{ and }B) = P(A)P(B)$ is correct for independent events.

$P(A\text{ or }B) = P(A) + P(B)$ is correct for mutually exclusive events.

So you were right about the first one.

More generally, there is an "inclusion–exclusion" rule like this

$P(A\text{ or }B) = P(A) + P(B) - P(A\text{ and } B)$ for any events $A, B$. They need not have a special relation like "mutually exclusive" or "independent".

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    $\begingroup$ Perhaps worth remarking that two independent events, both with positive probability, cannot be mutually exclusive. $\endgroup$ – paw88789 Mar 28 '17 at 5:18

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