Apples are sold in cartons of 25. David bought home 2 cartons an found 3 rotten apples in the first carton. There were thrice as many rotten apples in the second carton.
1. Find the probability of randomly selecting a good apple from carton 2.
2. If 2 apples are randomly selected from carton 1, find the probability that both are good.
3. Suppose that both cartons are covered up, and you're given the following options,
Option 1: randomly choose a box and then choose 2 apples from that box.
Option 2: randomly choose 1 apple from each box
Which option will you take in order to increase your chances of getting 2 good apples? Validate your choice.
He has 50 apples in total,3 rotten
50-3=47
3*3=9
47-9=38
22 apples in carton 1
16 apples in carton 2
1.16/25
2.22/25
3.option 2
because you have a higher probability of getting 2 good apples
To answer these questions, let's break it down step by step:
1. Probability of randomly selecting a good apple from carton 2:
First, we need to determine the number of rotten apples in the second carton. It is given in the question that there were thrice as many rotten apples in the second carton. Since there were 3 rotten apples in the first carton, this means there are 3 * 3 = 9 rotten apples in the second carton.
The total number of apples in the second carton is 25. Therefore, the number of good apples in the second carton is 25 - 9 = 16.
The probability of randomly selecting a good apple from the second carton is the ratio of the number of good apples to the total number of apples: 16/25 = 0.64 or 64%.
2. Probability that both apples selected from carton 1 are good:
In carton 1, there were 3 rotten apples. Therefore, the number of good apples in carton 1 is 25 - 3 = 22.
To calculate the probability, we need to determine how many ways we can select 2 apples from 22 good apples, out of a total of 25 apples.
Using the combination formula, the total number of ways to select 2 apples from 25 is C(25, 2) = 25! / (2! * (25 - 2)!) = 300.
The number of favorable outcomes (selecting 2 good apples) is C(22, 2) = 22! / (2! * (22 - 2)!) = 231.
Therefore, the probability that both apples selected from carton 1 are good is 231/300 = 0.77 or 77%.
3. Choosing between Option 1 and Option 2 to increase chances of selecting 2 good apples:
Option 1: Randomly choose a box and then choose 2 apples from that box.
For this option, we have two cases:
- If we randomly choose the first carton: The probability of selecting 2 good apples from carton 1, as calculated above, is 77%.
- If we randomly choose the second carton: The probability of selecting 2 good apples from carton 2 is the probability of selecting a good apple from carton 2, which we calculated to be 64%. Since the number of good apples in the second carton is 16, the probability of selecting the second good apple is 16/24 (assuming the first apple was good).
Option 2: Randomly choose 1 apple from each box.
In this case, we have two independent events. The first apple could be selected from either carton with equal probability. The probability of selecting a good apple from the first carton is 22/25, and from the second carton is 16/25.
To find the probability of both apples being good, we multiply the probabilities of selecting a good apple from each carton: (22/25) * (16/25) = 0.704 or 70.4%.
Based on the calculations, Option 1 (randomly choosing a box and then choosing 2 apples) gives a higher probability of selecting 2 good apples, with a probability of 77% compared to Option 2's probability of 70.4%. Therefore, Option 1 should be chosen in order to increase the chances of getting 2 good apples.