What is the probablity of January having 5 sundays. Similarly for the other months

Question

I'm trying to find out what is the probability that a randomly chosen January will have 5 Sundays. Of course the answer for 4 Sundays would be 1. I presume that 31 day months will have a higher probability of having 5 then 30 day months. Of course, February in a non-leap year has 0 probablity of having 5 sundays and in a leap year will have 5 only if 1st Feb is a Sunday. Therefore in a leap year P(Feb,5) = 1/7 and over a 400 year time period the P(Feb,5) will be 99/2800. I presume all 31 day months will have the same probablity which should be higher than 30 day months and in turn will be higher than 99/2800. I've worked out P(31d month,5) will be 223/343 and P(30d month,5) is 19/49. Is this right?

Answer 1

January will have 5 Sundays if January 1 falls on Friday, Saturday or Sunday. Therefore the probability is 3/7. You can apply the same logic for other months.

Answer 2

You have to be a little careful calculating this. With the leap year rules the calendar repeats after 400 years (the number of days in 400 years is divisible by $7$). In any period of 400 years using the current calendar, 1 January will fall on:

Sunday 58 times

Monday 56 times

Tuesday 58 times

Wednesday 57 times

Thursday 57 times

Friday 58 times

Saturday 56 times

Answer 3

now it is sure that the month of jan will have 4 sundays as you said. so 31-28=3 days remaining

the remaining 3 days can have different combination of days

sun-mon-tue

mon-tue-wed

tue-wed-thu

wed-thu-fri

thu-fri-sat

fri-sat-sun

sat-sun-mon

so there are total 7 combinations and 3 of these combinations contains sunday

so probability of having 5 sundays in the month of january is P(E)= 3/7