tsmid said:
The solution I suggested <i>is</i> the same problem, just scaled up by a factor 2, which leaves the probabilities unchanged (whether you have 1/1 solid/striped balls in the bag or 0.5/0.5 doesn't matter; you could also have 100/100, the probability would still be 50%).
If you scale the problem up by a factor of 2 (not a well defined operation), it becomes:
"If a bag contains 4 balls, 2 of which are solid, and 2 of which are either solid or striped (50% chance either way), what is the probability that the third and fourth balls picked are striped, given that the first two picked are solid?"
Generally if you double all the numbers in a probability problem, the resulting probabilities don't stay the same. The probability of picking 1 red ball from a bag containing 1 red and 1 blue ball is 1/2. The probability of picking 2 red balls from a bag containing 2 red and 2 blue balls is 1/4 (assuming the first ball picked is replaced before the second draw), or 1/6 (assuming no replacement).
You can also use the following argument:
if the second ball added to the bag is a striped one, the probability of drawing it is 100% if the ball first drawn is a solid one;
if the second ball added is a solid one, the probability of drawing a striped is 0% as there are no striped balls in the bag at all;
the average of these two possibilities is 50%.
Careful with this approach - you're twice as likely to pick (on the first draw) a solid ball from a bag containing two solid balls than if the bag contains one solid and one stiped ball, so the average should be weighed accordingly.
Suppose the bag contains 1 million balls, and you know that either they are all solid, or one is solid and the other 999,999 balls are striped (there's a 50% chance either way). If one ball is then picked and turns out to be solid, is the probability that the second one picked is striped still 1 in 2?
Are the two possibilities:
• "you picked one of the million solid balls in the bag on your first draw"
and
• "you picked the only solid ball out of a bag containing 1 million balls on your first draw"
equally likely?
I would recommend however in these kind of cases to always turn the probabilities into an integer number of objects (like in my first solution above); this makes the solution more intuitive and you avoid the pitfalls of working with formal probabilities alone; you just have to take care that you scale up all numbers by the same factor (e.g. when drawing balls like in this case).
Dinosaur did this on the first page of this thread:
Dinosaur said:
I can never remember the applicable formulae or the text book logic for such problems. I usually getr the correct answer by imagining it as a sampling problem.
Suppose you did this process 60 times (chosen due to being divisible by a lot of different factors).
* 30 times a solid ball was transferred, and 30 times you drew a solid ball from the bag.
* 30 times a striped ball was transferred. In 15 of the 30 cases, you expect to draw a solid ball the second time. In 15 cases, you expect to draw a striped ball.
From the above, a solid ball was drawn a total of 45 times: In 15 cases, a second draw would result in a striped ball. In 30 cases, it would result in a solid ball.
The above leads me to believe that the probablity of drawing a striped ball is 1/3.
