There are 2 white marbles in urn A and 4 red marbles in urn B . At each step of the process a marble is selected from each urn and the 2 marbles are interchanged.What is the probability that there 2 red marbles in in urn A after three steps....please give an explanation
3 answers
can I pls get the answer
Let the states be:
R0 = 0 red marbles in urn A
R1 = 1 red marble in urn A
R2 = 2 red marbles in urn A
Initially, urn A has 2 white marbles and B has 4 red marbles
R0 R1 R2
A(0) = [ 1 0 0 ]
TPM = [ [ 0 1 0 ],
[ 1/8 4/8 3/8 ],
[ 0 1/2 1/2 ] ]
Explanation:
State R0
Urn A = w w
Urn B = r r r r
Probability of moving to state R1 is 1.
State R1
Urn A = w r
Urn B = w r r r
Probability of moving to state R0 = 1/2 * 1/4 = 1/8
Probability of moving to state R2 = 1/2 * 3/4 = 3/8
Probability of staying in state R1 = 1/2 * 1/4 + 1/2 * 3/4 = 4/8 (Exchanging white with white and red with red)
State R2
Urn A = r r
Urn B = w w r r
P(moving to R0) = 0
P(moving to R1) = 1 * 2/4 = 0.5
P(staying in R2) = 1 * 2/4 = 0.5
After 3 steps:
A(3) = A(0) * P^3
R0 R1 R2
A(3) = [ 0.0625 0.5625 0.3750 ]
Probability that there are 2 red marbles in urn A after 3 steps = 0.5625
R0 = 0 red marbles in urn A
R1 = 1 red marble in urn A
R2 = 2 red marbles in urn A
Initially, urn A has 2 white marbles and B has 4 red marbles
R0 R1 R2
A(0) = [ 1 0 0 ]
TPM = [ [ 0 1 0 ],
[ 1/8 4/8 3/8 ],
[ 0 1/2 1/2 ] ]
Explanation:
State R0
Urn A = w w
Urn B = r r r r
Probability of moving to state R1 is 1.
State R1
Urn A = w r
Urn B = w r r r
Probability of moving to state R0 = 1/2 * 1/4 = 1/8
Probability of moving to state R2 = 1/2 * 3/4 = 3/8
Probability of staying in state R1 = 1/2 * 1/4 + 1/2 * 3/4 = 4/8 (Exchanging white with white and red with red)
State R2
Urn A = r r
Urn B = w w r r
P(moving to R0) = 0
P(moving to R1) = 1 * 2/4 = 0.5
P(staying in R2) = 1 * 2/4 = 0.5
After 3 steps:
A(3) = A(0) * P^3
R0 R1 R2
A(3) = [ 0.0625 0.5625 0.3750 ]
Probability that there are 2 red marbles in urn A after 3 steps = 0.5625
To find the probability that there are 2 red marbles in urn A after three steps, we can calculate the probability of being in state R2 after three steps.
To transition from state R1 to R2, we need to select another red marble from urn B. The probability of selecting a red marble from urn B is 3/4, since urn B initially contains 4 red marbles. The probability of staying in state R2 after moving from R1 is also 1/2, since after selecting a red marble from urn B, we need to select a red marble from urn A as well. So, the probability of moving from R1 to R2 and staying in R2 is (3/4) * (1/2) = 3/8.
To transition from state R2 to R2, we need to select two red marbles from urn B. The probability of selecting two red marbles from urn B is (2/4) * (1/3) = 1/6. Since we are already in state R2, the probability of staying in R2 is 1. So, the probability of moving from R2 to R2 and staying in R2 is (1/6) * 1 = 1/6.
To calculate the probability of being in state R2 after three steps, we need to multiply the probabilities of the different paths that lead to state R2. There are two paths: R1 to R2 and R2 to R2. So, the probability of being in state R2 after three steps is (3/8) + (1/6) = 19/48.
Therefore, the probability that there are 2 red marbles in urn A after three steps is 19/48.
To transition from state R1 to R2, we need to select another red marble from urn B. The probability of selecting a red marble from urn B is 3/4, since urn B initially contains 4 red marbles. The probability of staying in state R2 after moving from R1 is also 1/2, since after selecting a red marble from urn B, we need to select a red marble from urn A as well. So, the probability of moving from R1 to R2 and staying in R2 is (3/4) * (1/2) = 3/8.
To transition from state R2 to R2, we need to select two red marbles from urn B. The probability of selecting two red marbles from urn B is (2/4) * (1/3) = 1/6. Since we are already in state R2, the probability of staying in R2 is 1. So, the probability of moving from R2 to R2 and staying in R2 is (1/6) * 1 = 1/6.
To calculate the probability of being in state R2 after three steps, we need to multiply the probabilities of the different paths that lead to state R2. There are two paths: R1 to R2 and R2 to R2. So, the probability of being in state R2 after three steps is (3/8) + (1/6) = 19/48.
Therefore, the probability that there are 2 red marbles in urn A after three steps is 19/48.