
Dan D. answered 06/06/16
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The main concepts are that
(probability of something) = (# of outcomes for that something) / (number of total outcomes)
where we assume each outcome has the same chance.
and that
(probability A happens and that B happens) = (probability of A) * (probability of B)
provided that A and B are independent (one happening doesn't change how the other happens.)
Prob of a red marble = 4 reds / (4+7+5 marbles) = 4/16 = 1/4
put it back and then
Prob of blue marble = 7 blues / (4+7+5 marbles) = 7/16
These are independent so
Prob both happen is (1/4) * (7/16) = 7/64 ~ 0.109
In the second version we start with the same:
Prob of a red marble = 4 reds / (4+7+5 marbles) = 4/16 = 1/4
but we don't put it back so when we draw for a blue marble we have:
Prob of blue marble = 7 blues / (3+7+5 marbles) = 7/15
and again the prob for both is
Prob both happen is (1/4) * (7/15) = 7/60 ~ 0.117
Note that this second version has a bit higher probability because
with the one red marble removed there is a bit higher chance of getting a blue.