Relative And Absolute Probability (Probability Of Picking A Red Ball)

[Agent Smith]

TL;DR

Given $r$ red balls, $g$ green balls and $b$ blue balls,
$r > g, r > b$ AND $r < g + b$

A bag contains $4$ red balls, $3$ green balls and $2$ blue balls.
A random ball is selected from this bag.
P(ball is red) = $P(R) = \frac{4}{9}$
P(ball is green) = $P(G) = \frac{3}{9}$
P(ball is blue) = $P(B) = \frac{2}{9}$

P(ball is not red) = $P(\neg R) = \frac{3}{9} + \frac{2}{9} = \frac{5}{9}$

$P(\neg R)$ because $P(\neg R) > \frac{1}{2}$ implies that it's unlikely that a random ball is red. I consider this absolute probability.

From the above we can see that $P(R) >P(G) \wedge P(R) > P(B)$ and that means, given any random ball drawn from the bag, the ball is more likely to be red than green and is more likely to be red than blue. This I've treated as relative probability, as we're comparing colors.

Above is a diagram of the probability distribution of scenario. Figure A shows the probabilities of each individual color and Figure B clubs green and blue together. I suppose I don't have to tell readers that we have to look at the height of the columns for the red, green, and blue balls.

This might be elementary for most, but as a beginner I'm unable to answer the question "what is the probable color of a random ball drawn from the bag?" If I use Figure A, the answer is red, but if I use Figure B, the answer is not red, but either blue/green.

Can anyone help?

 

[FactChecker]

From the above we can see that $P(R) >P(G) \wedge P(R) > P(B)$

What does $P(G)\wedge P(R)$ mean?

This might be elementary for most, but as a beginner I'm unable to answer the question "what is the probable color of a random ball drawn from the bag?"

Good point. "what is the probable color" is a poorly phrased question. It would be better to ask "what is the most likely color".

If I use Figure A, the answer is red, but if I use Figure B, the answer is not red, but either blue/green.

With the poorly phrased question, I'm afraid that there is no good answer. With the better phrased question the answer is clearly "Red".

 

[Ibix]

This might be elementary for most, but as a beginner I'm unable to answer the question "what is the probable color of a random ball drawn from the bag?" If I use Figure A, the answer is red, but if I use Figure B, the answer is not red, but either blue/green.

Probability theory is very sensitive to the question you ask. If you ask "what is the most probable colour" the answer is "red". If you ask "is it more probable that I get red than some other colour" then the answer is "no".

A more extreme example would be to consider a thousand balls, 999 labelled uniquely with the numbers 1-999 and the thousandth one labelled 1. The most probable number to draw is 1, with a probability of 0.002. But you are far more likely to draw another number - 0.998.

So the question is: why do you want to know? If you have to bet on the colour of one draw, red is your best bet. If you have to bet on red-or-not for one draw, you're better to bet "not red".

 

[Agent Smith]

What does $P(G)\wedge P(R)$ mean?

Syntactically pristine would've been $(P(R) > P(G)) \wedge (P(R) > P(B))$
It means P(R) is greater than P(G) AND P(R) > P(B).

Good point. "what is the probable color" is a poorly phrased question. It would be better to ask "what is the most likely color".

Ok, what is the most likely color?
Which of the 2 probability distributions did/should you/we use?

Probability theory is very sensitive to the question you ask. If you ask "what is the most probable colour" the answer is "red". If you ask "is it more probable that I get red than some other colour" then the answer is "no".

I found that out, took me a whole day. So ... is the ball likely to be red? No. Is it likelier to be red than green? Yes. Is it likelier to be red than blue? Yes.

I thought of an interesting analogy. Let L = Death by lightning and A = Death by asteroid and S = Death by supernova.

$P(L) = \frac{4}{9}$ and $P(A) = \frac{3}{9}$ and $P(S) = \frac{2}{9}$

It is unlikely I'll die by lightning, and it's likelier that I'' die by asteroid or supernova. I find this extremely fascinating as it's likelier that I'll die by asteroid or supernova or both than by lightning. Lightning happens everyday; a supernova that can kill or asteroid heading towards our planet is happens once every billions/millions years

So the question is: why do you want to know? If you have to bet on the colour of one draw, red is your best bet. If you have to bet on red-or-not for one draw, you're better to bet "not red".

This is a good follow-up question. I shouldn't bet on red then, oui?

 

[Ibix]

So ... is the ball likely to be red? No.

That depends on whether you consider 4/9 "likely". If I'm drawing balls from a bucket, no I wouldn't. If you're talking my chances of surviving a swim across a crocodile-infested lake I'd consider it terrifyingly likely. There is no clear definition of "likely" in probability, although "more/less likely than..." is fine. So I would simply not use the work "likely".

Is it likelier to be red than green? Yes. Is it likelier to be red than blue? Yes.

Agreed.

It is unlikely I'll die by lightning, and it's likelier that I'' die by asteroid or supernova. I find this extremely fascinating as it's likelier that I'll die by asteroid or supernova or both than by lightning. Lightning happens everyday; a supernova that can kill or asteroid heading towards our planet is happens once every billions/millions years

I don't think this us a particularly helpful analogy because the probabilities don't match up. Deaths by lightning strike do occur (apparently 6 in the US so far this year. There has been one recorded death from meteorite strike anywhere, and that was in 1888. And your probability of death from supernova is zero, because there's nothing near enough and big enough to do the job. So you are vastly more likely to die from lightning strike than meteor and supernova combined, and still more hugely likely to die from something much more mundane.

This is a good follow-up question. I shouldn't bet on red then, oui?

Depends what the game is and what you are allowed to do - that's kind of the point. If you have to pick one colour, red is the best colour at p=4/9. If you're allowed to pick "not red", that's better at p=5/9 and if you can pick "not blue" that's even better at p=7/9.

 

[Agent Smith]

That depends on whether you consider 4/9 "likely". If I'm drawing balls from a bucket, no I wouldn't. If you're talking my chances of surviving a swim across a crocodile-infested lake I'd consider it terrifyingly likely. There is no clear definition of "likely" in probability, although "more/less likely than..." is fine. So I would simply not use the work "likely".

But "(more/Less)likely", "(more/less) unlikely", "likelier", "unlikelier", "(more/less) probable", "(more/less) improbable" are part of the vocabulary of probability theory.

What about 2 events that are equally "probable" at $\frac{1}{2}$? What's the correct/technical term for this probability scenario?

don't think this us a particularly helpful analogy because the probabilities don't match up.

I have to agree. I'm bad at examples. Could you find one that matches the scenario in the OP?

Depends what the game is and what you are allowed to do - that's kind of the point. If you have to pick one colour, red is the best colour at p=4/9. If you're allowed to pick "not red", that's better at p=5/9 and if you can pick "not blue" that's even better at p=7/9.

I see. Let's say the game is to correctly predict the color of a random ball from the bag. If your prediction is correct then you win, if it isn't then you lose. Would you bet on red? If yes can you show me the math why? If no, quare?

 

[Ibix]

But "(more/Less)likely", "(more/less) unlikely", "likelier", "unlikelier", "(more/less) probable", "(more/less) improbable" are part of the vocabulary of probability theory.

Sure, but "likely" on its own isn't, at least in my experience.

What about 2 events that are equally "probable" at $\frac{1}{2}$? What's the correct/technical term for this probability scenario?

Equally likely, equally probable, or equiprobable, I think.

Could you find one that matches the scenario in the OP?

Not off the top of my head. Card games and dice games are a rich source of such examples, though.

Let's say the game is to correctly predict the color of a random ball from the bag. If your prediction is correct then you win, if it isn't then you lose. Would you bet on red?

If I have to bet for some reason and I can only choose one colour, yes, because that's the most likely outcome so is my best chance of winning. It's not a good bet (I'll lose 5/9ths of the time) but it's the least bad option. I probably would not play if I had the choice, unless there was some very good reason (this game is rather like a charity raffle - your odds of winning aren't great, but the aim is to give you a small chance of reward in exchange for donating).

 

[Agent Smith]

Sure, but "likely" on its own isn't, at least in my experience.

"
"Either one is likely to trounce the rest of the competition."
— Michael Ordoña, Los Angeles Times, 18 Nov. 2023
From Merriam Webster Online Dictionary

Equally likely, equally probable, or equiprobable, I think.

. What does it mean to someone who's betting on (say) a coin-flip with 50-50 chances of heads?

Not off the top of my head. Card games and dice games are a rich source of such examples, though.

Ok.

If I have to bet for some reason and I can only choose one colour, yes, because that's the most likely outcome so is my best chance of winning. It's not a good bet (I'll lose 5/9ths of the time) but it's the least bad option. I probably would not play if I had the choice, unless there was some very good reason (this game is rather like a charity raffle - your odds of winning aren't great, but the aim is to give you a small chance of reward in exchange for donating).

P(losing) = 5/9. Correct!
Why is it "the least bad option"?
All raffles are the same, oui? P(win) is low or the odds of winning are low. Would you put that at less than 25% chance of winning?

 

[Ibix]

"Either one is likely to trounce the rest of the competition."
— Michael Ordoña, Los Angeles Times, 18 Nov. 2023

But can you find a reference in a statistics textbook? There's a difference between casual speech and technical language.

What does it mean to someone who's betting on (say) a coin-flip with 50-50 chances of heads?

It means they're equally likely to lose as they are to win. What else could it mean?

Why is it "the least bad option"?

Because if you pick any other colour you are less likely to win.

All raffles are the same, oui? P(win) is low or the odds of winning are low. Would you put that at less than 25% chance of winning?

This isn't really a point about probability theory - it's a point about behaviour. There isn't a straight answer to "would I bet on red" because it depends. Does it cost me something to play? What could I win? Is there a benefit to playing even of I lose? Can I choose not to play?

A raffle is an example of a game with terrible odds that I've got a very high probability of losing money on, but I play anyway because I want to give my money to the organisers. It isn't just the probability that matters to players. You can use probability theory to work out things like expected returns, but "would I bet on X" depends on many more factors than that.