Consider three sets \( E_1=\{1,2,3\} \), \( F_1=\{1,3,4\} \) and \( G_1=\{2,3,4,5\} \). Two elements are chosen at random, without replacement, from the set \( E_1 \) and let \( S_1 \) denote the set of these chosen elements. Let \( E_2=E_1-S_1 \) and \( F_2=F_1 \cup S_1 \). Now two elements are chosen at random, without replacement, from the set \( F_2 \) and let \( S_2 \) denote the set of these chosen elements.
Let \( G_2=G_1 \cup S_2 \). Finally, two elements are chosen at random, without replacement, from the set \( G_2 \) and let \( S_3 \) denote the set of these chosen elements.
Let \( E_3=E_2 \cup S_3 \). Given that \( E_1=E_3 \), let \( p \) be the conditional probability of the event \( S_1=\{1,2\} \). Then the value of \( p \) is:
