In new york, the probability of rain or snow during a winter day is 25%, for a s

Question

In new york, the probability of rain or snow during a winter day is 25%, for a spring day is 31%, for a summer day is 31%, and for a fall day is 23%. Each of these seasons lasts exactly one quarter of the year. Answer in proportions to two decimals (0.XX format).

part 1- What is Pr[ it (a random day) is winter ]?

part2- What is Pr[ it’s raining or snowing | it’s winter ]? (i.e., what is the probability of rain given a random winter day in new york?)

part 3- What is Pr[ it’s raining or snowing ]? (i.e., what is the probability that it rains on a random day in new york?)

part 4- What is Pr[ it’s winter | it’s raining or snowing ]? (i.e., what is the probability that it rains on a random day in new york?)

Explanation / Answer

1) Pr[ it (a random day) is winter) =0.25 (as each season last exactly one quarter)

2) Pr[ it’s raining or snowing | it’s winter ] =0.25

3)  Pr[ it’s raining or snowing ] =P(winter and rain or snow+spring and rain or snow+summer and rain or snow+fall and rain or snow)=0.25*0.25+0.25*0.31+0.25*0.31+0.25*0.23=0.275

4)Pr[ it’s winter | it’s raining or snowing ] =P(winter and rain or snow)/P( it’s raining or snowing )

=0.25*0.25/0.275=0.2273

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