2. Find the probability that a leap year selected at random, will contain 53 Mondays.
Answers
A normal year has 52 Mondays, 52 Tuesdays, 52 Wednesdays, 52 Thursdays, 52 Fridays, 52 Saturdays and 52 Sundays + 1 day that could be anything depending upon the year under consideration. In addition to this, a leap year has an extra day which might be a Monday or Tuesday or Wednesday...or Sunday.
We've now reduced the question to : what is the probability that in a given pair of consecutive days of the year one of them is a Monday?
Our sample space is S : {Monday-Tuesday, Tuesday-Wednesday, Wednesday-Thursday,..., Sunday-Monday}
Number of elements in S = n(S) = 7
What we want is a set A (say) that comprises of the elements Sunday-Monday and Monday-Tuesday i.e. A : {Sunday-Monday,Monday-Tuesday}
Number of elements in set A = n(A) = 2
By definition, probability of occurrence of A = n(A)/n(S) = 2/7
Therefore, probability that a leap year has 53 Mondays is 2/7. (Note that this is true for any day of the week, not just Monday)