Question

5 spades, 5 clubes, and 5 hearts

1. find P(Tk+l)in terms of P(Tk)

2. find P(T3)

3. find the prob the cards are together by characters after many cuts

Answer 1

The number of permutations of 5+5+5 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 .

Now there are possible cuts out of which 3 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 1/14). The events being disjoint,

1) So the recursive relation is

2) Use above recursion to find

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

0.08333333

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

PT_k <- function(k)
{
if(k==1)
{
return (3/25427001600)
}
else
{
return(PT_k(k-1)/7+1/14)
  
}
}

PT_k (89)