Can anyone explain it to me please?
What is the right solution?
In order to solve and better understand the above problem, I will start with a similar, but yet easier, problem (if you think it's too easy skip it).
Five marbles are in a bag: two are red and three are blue. If Andy randomly picks two of the marbles, what is the probability that at least one of the marbles he chooses will be red?
to solve this problem we need only to apply logic (no combinatorics or probability formulas needed !)
5 = total marbles
3 = blue marbles
2 = red marbles
Andy picks 2 marbles, and we want to know what is the probability that he will pick at least one red marble (note it can be two or one).
2/5 (first pick, probability of choosing red is two out of five)
1/4 (second pick, probability of choosing red is one out of four since one was removed in first pick).
Since the events are dependent we need multiply them.
2/5*1/4 = 2/20 = 1/10 (probability of choosing red in first pick and choosing red in second pick)
but as you can see thats not even one of the given choices !!
we need to add outcomes where Andy chooses only one red !
2/5*3/4 = 6/20 = 3/10 (choosing red in the first pick and choosing blue in the second pick)
3/5*2/4 = 6/20 = 3/10 (choosing blue in the first pick and choosing red in the second pick)
since those three events are independent we will add them one to another.
(3/10)+(3/10)+(1/10) = 7/10
the answer is then (C)
Moveing on to the posted question:
If you have 250 marbles, 15 are black, and you pull 10 at random, each day for 5 days (the marbles are then returned before the next selection), what is the probability that one of the black marbles will not be grabbed?
My opinion is that this is way too complex question to be in the GMAT. but nevertheless applying the same logic as in the first problem will help us solve this problem:
250 = total marbles
15 = black marbles
235 = not-black marbles (we really don't care what color).
the probablity of choosing a not-black marble in a single try is P1 = 235/250 = 94%
the probablity of choosing a not-black marble in ten tries is
P10 = (235/250)*(235/250)*(235/250)*(235/250)*(235/250)*(235/250)*(235/250)*(235/250)*(235/250)*(235/250) = (235/250)^10 = ~ 54%
choosing not-black in first,second,third... and so on pick, for a total of ten picks)
we will use (235/250) every pick (and not 234/250, 233/250) because the marbles are being returned after each selection (unlike the first problem).
Since we have the same process repeted over five days, and the events are dependent (statistically it dosen't matter if we pick 50 times in one day or pick 10 times a day for five days !).
we have to multiply (235/250) 50 times !!!
The outcome will then be (235/250)^50 = ~ 4.5%
we have ~ 4.5% of chooseing not-black every single time (meaning not choosing black marble) marble !