Q3.

A deck of playing cards contains 52 cards, 4 each with face values in the set {A, 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K}. Let us define face values of the top and bottom cards as X and Y. The shuffling algorithm is defined as follows :

Sh1 : Permute the cards randomly due to that each arrangement occurs with probability \(1/52!\)

Sh2 : If \(X \ne Y\), flip the biased coin that comes up heads with probability P and go back to Sh1 if heads turns up otherwise stop.

Each coin flip and each permutation is assumed to be independent of all the other randomization. What value of P will make X and Y independent random variables once this process of shuffling will stops ?

Q4.

Identify the number of distinct string of length 2 of the combinations of blue and yellow beads. The both ends of the strings are not marked due to this they are indistinguishable if interchanging the ends of one will results other. Denote \(b\) and \(y\) as a blue and yellow beads. Let us consider \(bb\), \(by\), \(yb\) and \(yy\) are the four different strings of length 2 when equivalence between strings is not taken into the consideration. The number of equivalent classes into which the set \(s = (bb, by, yb, yy)\) is divided by equivalent relation induced by the group of permutation \([\{\Pi_{1}, \Pi_{2}\}, 0]\) where :

\(\Pi_{1} = \begin{pmatrix} bb & by & yb & yy \\ bb & by & yb & yy \end{pmatrix}\)

\(\Pi_{2} = \begin{pmatrix} bb & by & yb & yy \\ bb & yb & by & yy \end{pmatrix}\)

Q5.

Design a K-Map for \(F(x, y, z) = x\overline{z} + xyz + y\overline{z}\) of \(F(x, y, z)\) and find prime implicants.