Each of k jars contains m white and n black balls. A ball is randomly chosen from jar 1 and transferred to jar 2, then a ball is randomly chosen from jar 2 and transferred to jar 3, etc. Finally, a ball is randomly chosen from jar k. Show that the probability that the last ball is white is the same as the probability that the first ball is white, i. e. , it is m/(m + n) .

Write a short Python program for problem 22. Provide indented code and comments explaining your solution.

#initial probability of getting a white ball

init_prob_white=15/25

#initial probability of getting a black ball

init_prob_black=10/25

#probability of getting a white ball given

#that we transferred a white ball in previous

#step is 16/26 which we represent by pww

pww=16/26

#probability of getting a white ball given

#that we transferred a black ball in previous

#step is 15/26 which we represent by pwb

pwb=15/26

#similarly

pbb=11/26;

pbw=10/26;

#The below two functions are recursive which calculate

#total probability right from the start

#The formula they use are

# Prob(W_k)=Prob(W_k-1)*Prob(W_k | W_k-1)+

# Prob(B_k-1)*Prob(W_k | B_k-1)

def pw(n):

if n==1:

return init_prob_white;

else:

return (((pww)*pw(n-1))+((pwb)*pb(n-1)))

def pb(n):

if n==1:

return 10/25;

else:

return (((pbb)*pb(n-1))+ ((pbw)*pw(n-1)))

#for k=5

k=5

x=pw(k)

print("After "+str(k)+" transfers probability="+str(x))

Explanation:

that without the operating systems you would have to control and synchronize computer hardware.