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I stumbled upon this question on the SAT test(Math Level 2) and I do not know how to solve.

I'd appreciate if someone will explain to me how to think about it as probability is not my strong suit.

  1. "A meteorologist reports that there is 30% probability of rain and no sun. If there is a 40% probability of no rain, then the probability of both rain and sun is

(A) 0.16 (B) 0.24 (C) 0.30 (D) 0.50 (E) 0.60

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    $\begingroup$ I would draw a Venn diagram if I were you. $\endgroup$ – Matthew Pilling Jan 28 at 21:42
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$P$(no rain$)\space=0.4 \implies P($rain$)=0.6$

$P$(rain and no sun)$\space =0.3$

$\implies P$(rain and sun)$\space =P$(rain)$\space- P($rain and no sun$) \space=0.6-0.3=0.3$

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There are a couple of steps involved, but the main things you need to consider are:

  1. The probability of event $A$ and the probability of event $A^c$ (the complement of $A$, or "the event that $A$ doesn't happen") sum to 1, i.e. $P(A) + P(A^c) = 1$. For example, the probability of "sun" and "no sun" add up to 1.

  2. For any two events $A$ and $B$, the probability of $A$ and $B$ and the probability of $A$ and $B^c$, sum to the probability of $A$, i.e. $P(A \cap B) + P(A \cap B^c) = P(A)$. For example, the probability of "rain and no sun" plus the probability of "rain and sun" add up to the probability of "rain".

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