A box always contains three marbles, each of which is green...

A box always contains three marbles, each of which is green or yellow. At regular intervals, one marble is selected at random from the box and removed, while another is put in its place according to the following rules: a green marble is replaced by a yellow marble with probability .3 (and otherwise by another green marble), while a yellow marble is equally likely to be replaced by either color. Let $X_{n}=$ the number of green marbles in the box after the $n$ th swap. $$ \begin{array}{l}{\text { (a) What are the possible values of } X_{n} ?} \\ {\text { (b) Construct the one-step transition matrix for this Markov chain. }} \\ {\text { (c) If all three marbles currently in the box are green, what is the probability the same will be }} \\ {\text { true three swaps from now? }}\end{array} $$ $$ \begin{array}{l}{\text { (d) If all three marbles currently in the box are green, what is the probability that the fourth }} \\ {\text { marble selected from the box will be green? [Hint: Use part (c). Be careful not to confuse the }} \\ {\text { color of the marble selected on the fourth swap with the color of the one that replaces it! }}\end{array} $$

A box always contains three marbles, each of which is green or yellow. At regular intervals, one
marble is selected at random from the box and removed, while another is put in its place
according to the following rules: a green marble is replaced by a yellow marble with probability .3 (and otherwise by another green marble), while a yellow marble is equally likely to be
replaced by either color. Let $X_{n}=$ the number of green marbles in the box after the $n$ th swap.
$$
\begin{array}{l}{\text { (a) What are the possible values of } X_{n} ?} \\ {\text { (b) Construct the one-step transition matrix for this Markov chain. }} \\ {\text { (c) If all three marbles currently in the box are green, what is the probability the same will be }} \\ {\text { true three swaps from now? }}\end{array}
$$
$$
\begin{array}{l}{\text { (d) If all three marbles currently in the box are green, what is the probability that the fourth }} \\ {\text { marble selected from the box will be green? [Hint: Use part (c). Be careful not to confuse the }} \\ {\text { color of the marble selected on the fourth swap with the color of the one that replaces it! }}\end{array}
$$

Key Concepts

Markov Chain

A Markov chain is a stochastic process characterized by the property that the future state of the process depends only on the current state and not on the sequence of events that preceded it. This memoryless property makes it particularly useful for modeling systems that evolve over time with probabilistic transitions.

State Space

The state space in a Markov chain is the set of all possible states that the process can occupy. In problems like this, where outcomes are defined by the number of a certain type of object (for example, the number of green marbles), the state space includes all count values that the process might assume.

One-step Transition Matrix

The one-step transition matrix is a table that outlines the probabilities of moving from one state to any other state in a single time step. It serves as a fundamental tool in Markov chain analysis, enabling the computation of transition probabilities for any number of steps using matrix operations.

Multistep Transition Probabilities

Multistep transition probabilities describe the likelihood of transitioning from one state to another over a defined number of steps. These can be derived by raising the one-step transition matrix to a power corresponding to the number of steps, thanks to the Markov property which ensures that the process’s future evolution depends only on its current state.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has occurred. In the context of stochastic processes and Markov chains, it is used to assess the likelihood of future events based on the current state of the system, such as predicting the state of the system after several transitions.

Transcript

00:01 So in this problem we are told that there is a box that always contains three marbles, which are either green or yellow, and then at every interval a marble is randomly selected.

00:12 And if a green marble is selected, it is replaced by a yellow marble with probability 0 .3.

00:19 So let's write this down as we go along.

00:22 So the probability we'll call this a yellow replacement, y sub r.

00:29 So the probability of replacing the randomly selected marble with yellow, given that it was a green one that was selected, is 0 .3.

00:42 And we're also told that if a yellow marble is randomly selected, then the probability of replacing it with a green marble is 0 .5.

00:51 So we can also say the probability of a green replacement, given that a yellow marble is selected, is 0 .5.

01:07 And the other thing we're told is that x sub n is the number of green marbles after n swaps or n replacements.

01:29 Part a asks what are the possible values for x sub n? so since the marbles can be replaced by either yellow or green, it's possible that you would have anywhere from zero to three green marbles in the box.

01:50 So these are the possible states for x sub n.

01:57 Then for b, we are asked to construct the one -step transition matrix.

02:06 So i'm just going to write the indices down first to guide us.

02:29 So there are quite a few calculations involved to calculate all of these.

02:33 Some things we know is you can't, since there's only one, there's always just one replacement for each stage, the number of green marbles can't jump by two.

02:43 So you can't go from zero to two or zero to three.

02:51 Likewise, you can't go from 3 to 0 or from 3 to 1, and you can't go from 2 to 0.

02:59 So let's calculate this one in detail.

03:05 So we'll do this one, and then i will let you do the rest of them yourself, and i'll just give you the entries, rather than doing each calculation individually.

03:14 It would take too long.

03:17 So that is the probability of going from one green marble to one green marble in one step.

03:30 Probability that x sub n plus 1 equals 1 given x sub n equals 1.

03:44 And so there's only two ways that this could happen.

03:47 You could have a green marble selected and replaced by a green marble, the one green marble selected and replaced by a green marble, or one of the two yellow marble selected and replaced by a yellow marble.

03:59 So we can write out the total probability.

04:02 So the probability of selecting a green marble times the probability of replacing it with a green marble, given that we've selected a green marble.

04:20 Plus the probability of selecting a yellow marble times the probability of replacing that yellow marble with another yellow one, given that we've selected a yellow one.

04:39 And so there's only one green marble in the box at this stage.

04:44 So the probability of selecting the green marble is 1 out of 3.

04:50 And the probability of replacing a green marble with a green marble is the complement of replacing it with a yellow marble.

04:58 So it's going to be 0 .7.

05:03 The probability of selecting a yellow marble is 2 out of 3, because 2 out of 3 of the marbles are yellow.

05:12 And the probability of replacing the yellow marble with another yellow marble is the complement of this, which is 0 .5.

05:27 And so this comes out to 0 .567.

05:38 So then we just repeat this process for each of these entries.

05:47 And i'll go ahead and do that behind the scenes and then just present the final numbers to you.

05:56 And here you have it.

05:57 Here's the one -step transition matrix for this markov chain.

06:10 And so for part c, we are asked if the current state is, we have three green marbles, so all marbles are green, what is the probability, the same will be true three swaps from now...

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