Question

A number is chosen randomly from one of the two sets, $A=\{1801,1802, \ldots . ., 1899,1900\} \&$ $B =\{1901,1902, \ldots \ldots, 1999,2000\}$. If the number chosen represents a calender year. If the probability that it has 53 Sundays is $\frac{ p }{1400}$, then find the value of $p$.

Answer 1

$A =\{1801,1802, \ldots ., 1899,1900\}$
$B =\{1901,1902, \ldots ., 1999,2000\}$

E: randomly chosen year has 53 sundays
$P(E) =P(E \cap L)+P(E \cap O) $
$ =P(L) \cdot P(E / L)+P(O) \cdot P(E / O) $
$ =\frac{1}{2}\left[\frac{24}{100} \cdot \frac{2}{7}+\frac{76}{100} \cdot \frac{1}{7}\right]+\frac{1}{2}\left[\frac{25}{100} \cdot \frac{2}{7}+\frac{75}{100} \cdot \frac{1}{7}\right] $
$=\frac{249}{1400}=\frac{p}{1400} \Rightarrow p=249$