We are given a basket containing 12 apples, of which 3 are rotten, and 9 are good. We are asked to find the probability of drawing at most one rotten apple when 3 apples are drawn at random.
Step 1:
The total number of ways to select 3 apples out of 12 is given by the combination formula:
(312)=3×2×112×11×10=220
Step 2:
We need to find the probability of drawing at most one rotten apple, which means there can be either 0 or 1 rotten apple in the selection of 3 apples.
Case 1: Drawing 0 rotten apples (all 3 apples are good):
The number of ways to select 3 good apples from 9 is:
(39)=3×2×19×8×7=84
Case 2: Drawing 1 rotten apple (2 good apples and 1 rotten apple):
The number of ways to select 1 rotten apple from 3 and 2 good apples from 9 is:
(13)×(29)=3×2×19×8=3×36=108
Step 3:
Now, the total number of favorable outcomes is the sum of the two cases:
84+108=192
Step 4:
The probability of drawing at most one rotten apple is the ratio of favorable outcomes to total outcomes:
P(at most one rotten apple)=220192=5548
Thus, the probability that at most one rotten apple is drawn is
5548.