Question 478952
the ticket problem assumes no replacement.
the probability of drawing the first odd ticket is 5/9.
once you draw that ticket, without replacement, then then there are 4 odd tickets left out of a total of 8, so the probability of getting an odd ticket on the second draw is 4/8.
5/9 * 4/8 = 20/72 = 5/18
5C2/9C2 = 10/36 = 5/18
they're the same.
with your card problem, there is no replacement.
using combinations, the answer you show is:
13C3/52C3 = 286/22100 = .012941176 
using probabilities, the answer would be 13/52 * 13/52 * 13/52 = .015625
the answer are not the same because replacement was assumed using the probabilties while no replacement was assumed when using the combination formulas.
that's the difference between each way.
the two different methods are comparable if you do not replace after every draw.
they are not comparable if you replace after every draw.
if you assumed no replacement in the card problem, then the probability would have been calculated as 13/52 * 12/51 * 11/50 = .012941176
the two methods are comparable in the card problem when you assume no replacement.
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if you look at 13C3 / 52C3 and work out the math, this is what you get:
{{{13C3 = 13! / (3! * 10!)}}}
{{{52C3 = 52! / (3! * 49!)}}}
13C3 / 52C3 becomes equal to:
{{{(13! / (3! * 10!)) / (52! / (3! * 49!))}}}
this is equivalent to:
{{{(13! / (3! * 10!)) * ((3! * 49!)/52!)}}}
this can be rewritten as:
{{{(13! * 3! * 49!) / (3! * 10! * 52!)}}}
going a little further, this can be rewritten as:
{{{(13 * 12 * 11 * 10! * 3! * 49!) / (3! * 10! * 52 * 51 * 50 * 49!)}}}
there's a 3! in the numerator and the denominator that cancels out.
there's a 10! in  the numerator and the denominator that cancels out.
there's a 49! in the numerator and the denominator that cancels out.
your are left with:
{{{(13 * 12 * 11) / (52 * 51 * 50)}}}
if you do the problem the alternate way using the probabilities for each event, then you get the same equation without replacement.
it is:
{{{(13/52) * (12/51) * ( 12/50)}}} which can be rewritten as:
{{{(13 * 12 * 11) / (52 * 51 * 50)}}}
bottom line:
the two methods are comparable if you assume no replacement.
they are not comparable if  you assume replacement.