So this might be a stupid question but here goes: I thought I’ll choose numbers here so I chose the probability of rain tomorrow will be 0.9 and probability of no rain tomorrow should be 0.1 (1-0.9). The question is asking for max value of AB i.e. max value that it will rain and it will not rain tomorrow? How can it rain AND not rain tomorrow? Aren’t those mutually exclusive events? So I chose (A).

I’m pretty sure I am doing something stupid here so thanks in advance for helping me out!

A and B are probabilities. So in your example, AB would be 0.9 \times 0.1 = 0.09.

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Yeah but AB means probability of A AND B which means “what is the probability that it will rain tomorrow AND it won’t rain tomorrow?”. But how does that make sense? How can it rain AND not rain tomorrow?

A \times B is not A \cap B. The former is asking you to calculate the products of two probabilities, the latter is asking you to find the probability that A and B can both happen. As you noted, that’s impossible, and hence A \cap B = 0, but that is not what the question is asking.

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Here there is two total event and that makes 2 choices.

Rain and not rain

so

A is probability of rain tomorrow that is 1/2

B is probability of not rain tomorrow that is 1/2

now you can easily get maximun value of A*B =

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When you want to maximize a product/sum, I have learned that there are 2 ways: One is if you look at extremes and another is when the two parts are equal (like how regular shapes are in geometry).

Given you are dealing with fractions, good to look at both. So now, for eg, if A is 0.5 and B is 0.5, then AxB will be 0.25… and if A is 0.9 and B is 0.1, then AxB will be 0.09. It is impossible for AB to be bigger than 0.25, so we choose B.

Now logically, yh, you are right… it should be 0 because it is impossible for mutually independent events to happen simultaneously, so your question is not stupid : )

But is this Q asking you what is the probability of events A and B to happen together? No, right? It is asking you what is max value of the **product** of Probability of event A and Probability of event B. So you just multiply and get the product.

And this question seems a bit bleh, imo. Is this ETS?

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Ah okay. So if the question asked to find the probability of A and B and if the events are independent then prob of A and B = AB. If they are mutually exclusive, then 0. Here, they are just asking about the product of A x B.

Thank you so much for always answering my quant queries!