A customer at Cavallaro's Fruit Stand picks a sample of 4 oranges at random from a crate containing 50 oranges, of which 4 are rotten. What is the probability that the sample contains 1 or more rotten oranges?
To find the probability that the sample contains 1 or more rotten oranges, we need to calculate the probability that none of the four oranges picked is rotten and subtract it from 1.
The probability of picking a rotten orange from the crate is 4/50. Therefore, the probability of picking a non-rotten orange is 1 - (4/50) = 46/50.
We need to calculate the probability of picking 4 non-rotten oranges in a row. Since each pick is independent, we can multiply the probabilities.
So, the probability of picking 4 non-rotten oranges in a row is (46/50) * (46/50) * (46/50) * (46/50).
To find the probability that the sample contains 1 or more rotten oranges, we subtract this probability from 1.
Therefore, the probability that the sample contains 1 or more rotten oranges is 1 - [(46/50) * (46/50) * (46/50) * (46/50)].
To find the probability that the sample contains 1 or more rotten oranges, we need to calculate the complement of the probability that the sample contains no rotten oranges.
Step 1: Determine the total number of favorable outcomes.
The number of ways to choose 4 oranges from 50 is given by the combination formula: C(50, 4) = 50! / (4! * (50 - 4)!).
The number of ways to choose 4 non-rotten oranges from the 46 non-rotten oranges in the crate is given by the combination formula: C(46, 4) = 46! / (4! * (46 - 4)!).
Step 2: Determine the total number of possible outcomes.
The number of ways to choose 4 oranges from 50 is given by the combination formula: C(50, 4) = 50! / (4! * (50 - 4)!).
Step 3: Calculate the probability of the complement event.
The probability that the sample contains no rotten oranges is the number of favorable outcomes divided by the number of possible outcomes: C(46, 4) / C(50, 4).
Step 4: Calculate the probability of the desired event.
The probability that the sample contains 1 or more rotten oranges is equal to 1 minus the probability of the complement event: 1 - (C(46, 4) / C(50, 4)).
So the final answer is 1 - (C(46, 4) / C(50, 4)).