Question

A number is chosen randomly from one of the two sets, $A=\{1801,1802,\dots ..,1899,1900\}\&$ $B=\{1901,1902,\dots \dots ,1999,2000\}$. 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

Answer: 0249

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

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