# Total probability theorem or Bayes theorem?

How do I solve this problem??

The probability of receiving more than 300 mm of rain in each month is given in the following table. If a monthly rainfall record selected at random is found to have more than 300 mm of rain, what is the probability the record is for July? for December?

Jan 0.02 Feb 0.05 Mar 0.10 Apr 0.40 May 0.60 Jun 0.75 Jul 0.80 Aug 0.70 Sep 0.50 Oct 0.20 Nov 0.05 Dec 0.02

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Let $R$ be the event that in the month selected we have more than $300$ mm of rain, and let $J$ be the event the selected month is July. We want $\Pr(J|R)$. We have $$\Pr(J|R)=\frac{\Pr(J\cap R)}{\Pr(R)}.$$
Calculate. We have $\Pr(J\cap R)=\frac{1}{12}(0.8)$.
To calculate $\Pr(R)$, multiply $\frac{1}{12}$ by the probability of rain in each month, and add up.
Note that we have the multiplier $\frac{1}{12}$ on top, and in each term at the bottom. So they cancel, and our answer is just $0.8$ divided by the sum of the $12$ given probabilities.