Please help me with this probability problem

Lunaxa

I got this probability question as homework, but I got very confused as I didn't know if I should calculate the balls in the 2 boxes as whole or if I should calculate the probability of the first box then add the probability of the second. Any help is appreciated.
Thanks.


Here is the question:
There are 2 boxes, box A contains 20 white balls and 20 black balls, box B contains 10 white balls and 5 black balls. A box is chosen at random then a ball is taken at random from the chosen box, what is the probability that the ball is white?

First I thought, it should be the probability of box A + the probability of box B;
20/40 + 10/15 = 7/6
but then I thought it doesn't really make sense to me as this tells that I will always get a white ball?!

So I thought, I could do the probability of the whole lot;
30/55 = 6/11
this answer seems more reasonable, but then I was confused, as there are 2 boxes, so that the chances of a white ball being taken out of each ball should be different?

Yakuza

It's 0.5 * 20 / 40 + 0.5 * 10/15 or 0.58333 to 5 decimal places, (it's 7/12 as a fraction)
To get a white ball, you either have to get one from A or B.
To get one from A, you have to choose A (probability 1/2) then choose a white ball (probability 20 / 40), or 0.25000
To get one from B, you have to choose B (probability 1/2) then choose a white ball (probability 10 / 15), or 0.33333...
Add these together to get the total probability of getting a white ball.

Similarly, to get a black ball its 0.5 * 20 / 40 + 0.5 * 5/15 or 0.25 + 0.166667, .ie .416667

Both probabilities add to 1 which is a good sign

The question is asking two things : independent events and mutually exclusive events.
Choosing a box is independent from picking a white ball, so to get the probability of both happening (the "joint" probability), you have to multiply their respective probabilities to get the probability of both happening (i.e. 1/2 * 20/40).

To work out the prob. of two mutually exclusive events (things that both can't happen at the same time), you just add the probabilities together (in the context of this question, you can't choose a ball from box A and box B at the same time, so you add the respective (joint) probabilities together.

Your second answer would be correct if you emptied the contents of both boxes into a bigger box and pulled one ball at random from it!

Your first answer needs to weight the probability by the chances of choosing A or B (i.e multiply each term in it by a half).

(Feel free ignore the next line, it's some extra information)
If, for example, you threw a dice to see what box you chose from (only choosing box B if you get a six with the dice), then the answer above would have the "1/2"s replaced with 5/6 and 1/6 respectively (ignore this bit if I'm confusing you!)

Lunaxa

Yakuza wrote: »

It's 0.5 * 20 / 40 + 0.5 * 10/15 or 0.58333 to 5 decimal places, (it's 7/12 as a fraction)
To get a white ball, you either have to get one from A or B.
To get one from A, you have to choose A (probability 1/2) then choose a white ball (probability 20 / 40), or 0.25000
To get one from B, you have to choose B (probability 1/2) then choose a white ball (probability 10 / 15), or 0.33333...
Add these together to get the total probability of getting a white ball.

Similarly, to get a black ball its 0.5 * 20 / 40 + 0.5 * 5/15 or 0.25 + 0.166667, .ie .416667

Both probabilities add to 1 which is a good sign

The question is asking two things : independent events and mutually exclusive events.
Choosing a box is independent from picking a white ball, so to get the probability of both happening (the "joint" probability), you have to multiply their respective probabilities to get the probability of both happening (i.e. 1/2 * 20/40).

To work out the prob. of two mutually exclusive events (things that both can't happen at the same time), you just add the probabilities together (in the context of this question, you can't choose a ball from box A and box B at the same time, so you add the respective (joint) probabilities together.

Your second answer would be correct if you emptied the contents of both boxes into a bigger box and pulled one ball at random from it!

Your first answer needs to weight the probability by the chances of choosing A or B (i.e multiply each term in it by a half).

(Feel free ignore the next line, it's some extra information)
If, for example, you threw a dice to see what box you chose from (only choosing box B if you get a six with the dice), then the answer above would have the "1/2"s replaced with 5/6 and 1/6 respectively (ignore this bit if I'm confusing you!)

Thank you very much!!
You are so good at explaining!!
I understand it now!
Thanks again!

timsey tiger

Hi Yakuza

"The question is asking two things : independent events and mutually exclusive events.

Choosing a box is independent from picking a white ball, so to get the probability of both happening (the "joint" probability), you have to multiply their respective probabilities to get the probability of both happening (i.e. 1/2 * 20/40)."

Your solution is fine, however the explanation is not.

Picking the box and picking a white ball are not independent. For them to be independent the probability of picking a white ball would not change on the outcome of picking a box.

To get the probability of picking a white ball from box A, you multiplied the conditional probability of a white ball given you are picking from box A by the probability of picking box A.

A tree diagram is a good way of representing these types of questions.

Yakuza

My phrasing may not have been 100% correct, but I believe I got the point across.

Picking a box and picking a ball are independent events. (I agree, I shouldn't have said "Choosing a box is independent from picking a white ball" but "Choosing a box is independent from picking a ball" instead.

timsey tiger

Hi Yakuza

It's not your phrasing "picking a box and picking a ball are independent events" is still incorrect, because they are just not independent.

Knowledge of which box is picked changes the probability of picking a white ball.

Independence has a very specific meaning in terms of probability and it is not used in the problem above so it would be better to stop digging now.

Toddy1959

I never got to the hard sums myself but I remember Jimmy Carr (a noted maths genius!!!) on 8 out of 10 cats does countdown explaining that the probability of winning the Lottery was 50% - you either win or you don't :-) - I expect that he would have concluded that the probability of picking a white ball was 50% also.