Question

show coding and answer using R

A deck of 16 cards, from the bottom up, consists of 4 ♣, 4 ♠, 4 ♦, and 4 ♥. The cards are cut at a random place. Let Tk be the event that the all the cards of every suit are together after cut k.

(a) Assuming that all possible cuts are equally likely, find P(T2).

(b) Find P(Tk+1) in terms of P(Tk). (c) If the deck is repeatedly cut many, many times, what is the approximate probability the cards will be together by suit?

Answer 1

The number of permutations of 16 cards is .

Let the probability that the present position of cards is such that all the cards of every suit are together is

. All all the cards of every suit are together in ways,

So .

a)Now there are 16 possible cuts out of which 4 will results in all the suits together. So,

If the first cut does not results in all the suits together, the second cut can result in all the cards together with probability (if cut at the position you previously cut , its probability being 1/16). The events being disjoint,

b) So the recursive relation is

c) The recusrive relation is implemented in R. The probability converges to

• 0.07692308.

The R code for calculating the probability recursively is given below:

PT_k <- function(k)
{
if(k==1)
{
    return (1/36324288000/4)
}
else
{
    return(3*PT_k(k-1)/16+1/16)
  
}
}

PT_k (89)