A challenging math problem

[Guest sbe70]

Here is a probability mathematical problem that I think it will make quite a few math brain cells to work overtime.......

 

PROBLEM:

John and Mary has each one hat filled with numbered balls. The numbers are from 1 to 40 and there are no balls with the same number. Hence, there are 80 balls in total. John will pick two balls and Mary will pick 3 balls.

 

The process is that John pick one ball first. John looks at the number on the picked ball and writes it down. He then reach into the hat and removes the two balls that has the number just below and above the picked ball. Hence, after John picked the first ball and have removed the adjacent numbered balls the hat contains either 37 or 38 balls (if John had picked ball 1 or 40 there would only be one adjacent numbered ball instead of two). Now John repeats the process and picks one ball out of the 37/38 balls, writes its' number down and removes the one/two adjacent numbered balls again.

 

Now it is Mary's turn. She does exactly the same process as John with her hat but 3 times instead of 2.

 

Question 1: What is the likelihood that Mary picks one same ball as John?

 

Question 2: What is the likelihood that Mary picks two same balls as John?

 

Question 3: What is the expected number of same balls that Mary and John will pick (this number could be a non integer and less than 1)?

 

Question 4: What does a general formula to calculate question 3 look like if the number of balls and the number of picked balls by John and Mary are general parameters (e.g. parameters A,B,C)?

 

Question 5: If John and Mary has balls that has adjacent numbers the "hit value" is valued to 0.5. If John and Mary has balls that has same numbers the "hit value" is valued to 1. What is the expected number of same balls that Mary and John will pick (this number could be a non integer and less than 1)?

 

Question 6: What does a general formula to calculate question 5 look like if the number of balls,the number of picked balls by John and Mary and the adjacent "hit value" are general parameters (e.g. parameters A,B,C,D)?

[Guest maxscud]

Question 1: What is the likelihood that Mary picks one same ball as John?

 

does this question means the ball john picks for the first time, or the adjacent balls he picked are also included?