Answer:
a) 0.025 (or 0.0250)
Explanation:
a probability is always
desired cases / totally possible cases
"desired" means the cases for which we want to get the probabilty.
so, we have 16 apples.
2 of them are rotten.
when pulling just one apple, the probability for this apple to be rotten is
2/16 = 1/8 = 0.125
the probability that it is not rotten is
14/16 = 7/8 = 0.875 = 1 - 0.125
after pulling the first apple, we have only 15 asked left to pull the second. and then 14 to pull the third.
how want of them are rotten ? that depends on the result of the previous pulls.
if the first apple was good, we still have 2 rotten apples in the basket.
if not, we have only 1 left.
and so on.
all in all, we have 3 different desired situations as possible outcomes :
1. the first 2 pulls get the 2 rotten apples, the third one is per default good.
2. the first and the third pulls get the rotten apples. the second is good.
3. the second and the third pull get the rotten apples, the first is good.
ad 1)
the probabilty to get rotten apples on the first 2 pulls (and then 1 his apple) is
2/16 × 1/15 × 14/14 = 1/8 × 1/15 = 1/120 = 0.008333333...
ad 2)
the probabilty to a rotten apple on the first and then on the third pull (and a good apple on the second) is
2/16 × 14/15 × 1/14 = 1/8 × 1/15 = 1/120 = 0.008333333...
ad 3)
the probabilty to pull first a good apple and then 2 rotten ones is
14/16 × 2/15 × 1/14 = 1/16 × 2/15 = 1/8 × 1/15 = ... = 0.008333333...
all 3 outcomes are completely independent outcomes. and their probabilities can be added for the probability of the total event :
3 × 1/120 = 3/120 = 1/40 = 0.025
so, a) is the correct answer.
as plausibility check, let's just think about the various answer options: to pull a rotten apple is by far less likely than to pull a good apple.
the chance to pick both rotten apples in just 3 attempts must be very, very small. so, it can't be any of the other rather large probability numbers.
also, another check that 1/120 is the correct probability to pick both rotten apples : there are 16 over 2 possible combinations to pick 2 spread out of 16. and only one of these combinations (both rotten apples) is desired.
16 over 2 combinations is
16! / (2! × (16 - 2)!) = 16×15/2 = 8×15 = 120
the probability for that 1 combination is then
1/120.