The probability that a leap year selected at random contains either 53 Sundays or 53 Mondays, is

$\frac{3}{7}$

A leap year consists of 366 days comprising if 52 weeks and 2 days. These are 7 possibilities for these 2 extra days viz., (i) Sundays, Monday (ii) Monday, Tuesday (iii) Tuesday, Wednesday (iv) Wednesday, Thursday (v) Thursday, Friday (vi) Friday, Saturday (vii) Saturday, Sunday.

Let us consider two events:

A : The leap year contains 53 Sundays

B : The leap year contains 53 Mondays.

We have,

$P(A) =\frac{2}{7}, P(B) =\frac{2}{7}, P(A ∩ B)=\frac{1}{7}$

∴ Required probability

$= P(A ∪ B)$

$= P(A) + P(B) - P(A ∩ B) = \frac{2}{7} +\frac{2}{7} -\frac{1}{7}=\frac{3}{7}$