Probability Question?

Question

If a randomly selected month of a randomly selected year has five sundays, what is the probability that it s the month of July ?

[Note: Use correct definition of a leap year. Every fourth year is not a leap year. ]

Answer 1

Here’s what I think is the shortest method to get the answer. It doesn’t really matter if the day is Sunday since each day of the week will have the same probability over a long period of years.

Over a long time, each month will be featured in the list of months with 5 Sundays. The question is how often a given month will feature. This probability is related to the number of extra days (above 28) in that month.

For a non-leap year, there are 29 total extra days in the year, 3 of them are in July. For leap years, there are 30 extra days. In a full cycle of 2800 years, there are 679 leap years, so on average there are 29 + 679/2800 extra days = 29 + 97/400.

Required probability = 3 / (29 + 97/400) = 400/3899

Note that if we worked with one-in-four leap years, we would get 4/39, which has a 0.026% error.

For months with 31 days, P31 = 400/3899 = 1200/11697
For months with 30 days, P30 = 800/11697
For February, Pfeb = 97/11697
Notice that 7*P31 + 4*P30 + Pfeb = 1

Answer 2

for july and others months with 31 days, only if the first or the second or the third day is sunday, there is 5 sundays. Then 3 possibilities.

For april and others months with 30 days, only two possibilities.

For february with 28 days, no possibility to have 5 sundays.

Then, there are (from january to december) :

3 + 0 + 3 + 2 + 3 + 2 + 3 + 3 + 2 + 3 + 2 + 3 = 29 possibilities.

The probability that it is july is 3/29

No ?

For a leap year, the probability would be 3/30

Answer 3

We are going to consider four different kinds of years: Before Leap, Leap, After Leap, Mid Leap (i.e. 2 years after leap.)

Assume that a month has 31 days. There are three ways that we could have five sundays - the first falls on Fri, Sat, or Sun. There are seven months with 31 days and all seven of these months fall on all four kinds of years. So we 3*7*4 = 84 ways of getting this.

Assume a month a 30 days. There are two Five Sunday options - The first falls on Sat or Sun. There are five, 30 day months and all of them fall in all four years. This gives us 2*7*4 = 56 way.

Assume its a February non-leap. There is no way to get five sundays out of a 28 day month.

Assume it a February leap. The only way to get five sundays is if the first falls on a Sunday.

Our total number of five sunday months is 84+56+1 = 141.

Of these 141 months, four of them are July (July Leap, July before Leap, July after leap, July mid leap).

p = 4/141

Answer 4

Madhukar you provided some real intelligent help to my question in the engineering section (your answer was correct) but how did you get the ".001" in your calculations?

the .001 you added to the i
and how did you get the 20 you divided by the 112000?

Can you explain how you worked it out related to the circuit.
Sorry for posting here...there was no how i could reach you.

Thanks in advance