# Probability of the last ball picked from a box is blue

In a box there are $$R$$ red balls and $$B$$ blue balls. $$R$$ and $$B$$ are even numbers, and $$R>B$$.

Initially a random ball is drawn from the box, we take a look on its color and keep the ball out of the box. After, we keep drawning balls from the box until we get a ball where its color is different from the first one. When it happens, we place this ball(the one with different color) back in the box and the process is restarted until there is no balls left.

The question is, what is the probability of the last ball picked from the ball is blue?

EDIT:
What I tried to do was, solve the problem backwards, that means that for the blue ball be the last one the drawns in the box must be specific. But I got nowhere with this idea.

The other thing that I tried was to write all possibilities with $$R=4$$ and $$B=2$$, but I don't know how to deal with the restart of the process.

Someone has any ideia how to model this problem?

EDIT 2:
Here is an example:
Let's say that we have $$50$$ red balls and $$14$$ blue balls.

Suppose that the firts drawn ball is red, in a such a way that now there are $$49$$ balls and $$14$$ blue balls inside the box. The second drawn is a red ball too, so now we have $$48$$ red balls and $$14$$ blue balls inside the box. Let's say that in the third drawn the ball is blue, as the last ball is a different color from the others taken out of the box, the blue ball is put back in the box and the process restarts with $$48$$ red balls and 14 blue balls. This keep going until there is no balls left.

• Welcome to Mathematics Stack Exchange! As this site is meant to be a useful repository rather than a Do My Homework forum, it's common courtesy to show what you've already tried, and really narrow down what you're struggling with. Most people here are glad to help once you've adequately motivated the problem. Quick Guide to attracting answers and preventing your question from being deleted. Good luck! Aug 7, 2022 at 2:37
• The source of the problem is from a college list of exercises. I didn't show any calculations because I don't really know where to start. So i posted it here to see if someone have a good ideia on this problem. I'm sorry for that. Aug 7, 2022 at 3:25
• Guys, I didn't want to cause any discussion here. I editted the question to show what i thought before posting here, but i honestly don't know if this is enough. Aug 7, 2022 at 12:40
• @QuantJammas: to come down to brass tacks, "When it happens, we place this ball(the one with different color) back in the box and the process is restarted" How is the ball put back, just at the point in the line from which picked, or randomly ? In other words, suppose I change from red to blue, will I restart from blue, or will it be random ? Aug 7, 2022 at 13:39
• @trueblueanil It will be random, so the process continues until there is no ball left. Aug 7, 2022 at 13:57

## 3 Answers

Let the desired probability be $$P_{R,B}$$ for $$R$$ red balls and $$B$$ blue balls. We characterize the process into 2 steps:

1. A ball is chosen at random and discarded.
2. A 2nd ball is chosen. If it has the same colour as the first ball, we discard it. If it has a different colour as the first ball, we replace it into the box and return to step 1.

Let's complete this cycle once, so that we do both steps and return to step 1. There are 3 mutually exclusive and exhaustive events (one of them HAS to occur):

1. All red balls have been discarded. Let's call this event $$\mathcal{A}$$. Clearly $$P(\mathcal{A})=\frac{R!B!}{(R+B)!}$$.
2. All blue balls have been discarded. Let's call this event $$\mathcal{B}$$. Clearly $$P(\mathcal{B})=\frac{R!B!}{(R+B)!}$$.
3. The box still contains at least 1 red and 1 blue ball. Let's call this event $$\mathcal{C}$$. Due to mutual exclusivity and exhaustivity, $$P(\mathcal{C})=1-\frac{2R!B!}{(R+B)!}$$

Our first line of attack in these types of problems should always be inductive. It turns out, in this case, we prepare strong induction on $$k=R+B$$, the total number of balls. We have $$R,B>0$$. Let our base case be $$k=2$$, so $$R=B=1$$. Clearly, due to symmetry, the probability $$P_{1,1}=\frac{1}{2}$$. To simplify the notation, let $$P_{R,B}=P_{k}$$ when $$k=R+B$$.

Inductive hypothesis: Let's assume $$P_{k'}=\frac{1}{2}$$ for all $$2\leq k'\leq k$$. Then, what is $$P_{k+1}$$? We can do both steps once and return to step 1. Then, we can condition on the exhaustive events described previously: If the 3rd event is the case, our inductive hypothesis guarantees the probability to be $$1/2$$. In the first 2 cases the probabilities of a blue ball being last are $$1$$ and $$0$$, respectively.

$$P_{k+1}=\frac{1}{2}\cdot P(\mathcal{C})+0\cdot P(\mathcal{B})+ 1\cdot P(\mathcal{A})$$

$$=\frac{1}{2}\cdot \left(1-\frac{2R!B!}{(R+B)!}\right)+ 1\cdot \frac{R!B!}{(R+B)!}=\frac{1}{2}$$

This completes our induction. The probability is ALWAYS $$1/2$$ for the blue ball to end up last. It doesn't matter how many total balls there are, nor what the relative proportion of red and blue balls are. It also doesn't matter whether $$R,B$$ are even or odd.

• Can you explain with more details how you calculated P(A),P(B) and P(C) please? Aug 8, 2022 at 17:03
• Any of $\binom{R+B}{R}$ possibilities are equally likely to be the first $R$ balls drawn. In only one of those cases are all $R$ balls red. So $P(\mathcal{A})=1/\binom{R+B}{R}=\frac{R!(R+B-R)!}{(R+B)!}=\frac{R!B!}{(R+B)!}$. Similarly, only one case corresponds to the first $B$ balls all being blue, so $P(\mathcal{B})=\frac{R!B!}{(R+B)!}$. Since one of $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$ HAS to occur, $P(\mathcal{A})+P(\mathcal{B})+P(\mathcal{C})=1$, so $P(\mathcal{C})=1-\frac{2R!B!}{(R+B)!}$. Aug 8, 2022 at 18:38

The "restart" of the process with $$R$$ red balls and $$B$$ blue balls remaining means that you have the same chance to draw blue last as if you started the entire process with $$R$$ red balls and $$B$$ blue balls.

Call this state of the box $$(R,B,i)$$ where the $$i$$ stands for "initial state".

What I think you need to think about actually are the cases that are not a "restart". There are two such cases for each possible description of the remaining contents of the box:

• You removed a red ball and did not replace it, leaving $$R$$ red balls and $$B$$ blue balls in the box. Call this state $$(R,B,r)$$ where the $$r$$ stands for "red".

• You removed a blue ball and did not replace it, leaving $$R$$ red balls and $$B$$ blue balls in the box. Call this state $$(R,B,b)$$ where the $$r$$ stands for "blue".

Notice that if $$R \geq 1$$ and $$B \geq 1$$, from $$(R,B,i)$$ you have probability $$R/(R+B)$$ to draw red and transition to $$(R-1,B,r)$$ and probability $$B/(R+B)$$ to draw blue and transition to $$(R,B-1,b).$$

If $$R \geq 1$$ and $$B \geq 1$$, from $$(R,B,r)$$ you have probability $$R/(R+B)$$ to draw red and transition to $$(R-1,B,r)$$ and probability $$B/(R+B)$$ to draw blue and transition to $$(R,B,i)$$ because you put the blue ball back and "restart".

Notice that if $$R \geq 1$$ and $$B \geq 1$$, from $$(R,B,i)$$ you have probability $$R/(R+B)$$ to draw red and transition to $$(R-1,B,r)$$ and probability $$B/(R+B)$$ to draw blue and transition to $$(R,B-1,b).$$

This will not give you the answer instantly, but if you look at the probabilities for each of the $$(R,B,i)$$ cases that you develop, you may see a pattern emerging, and this could lead to a more general solution. (It's easier to prove something once you know what it is you think you should prove.)

On some reflection, we can analyze as follows:

Consider "chunks" of red and blue, starting red, say. The game "consumes" the starting "chunk" of red but places the starting blue randomly, which may or may not create an additional chunk, eg if the blue is placed anywhere between the start and end of a blue chunk, no extra chunk is created, else an extra chunk is created.

This process will slowly result in a reduction of the number of chunks, so that ultimately a red-blue or blue-red chunk is left, and whichever chunk is last will produce the last ball.

Regardless of the arrangements at start, the game thus inevitably simplifies at the end to two chunks of red-blue or blue-red, and by symmetry, P(last ball is blue) = $$\large\frac12$$