How to Ensure Drawing at Least Two Yellow Candies from a Box
Imagine a box filled with candies of different colors. The box contains 5 green candies, 7 red candies, 15 blue candies, and 20 yellow candies. If someone is taking candies from the box randomly, the question often arises: how many candies must be taken to ensure at least two of them are yellow?
Using the Worst-Case Scenario Approach
To solve this problem systematically, we can use the worst-case scenario approach. This method involves considering the most unfavorable situation first to guarantee a positive outcome. Let's break down the problem step by step.
Count of Each Candy Color
Green: 5 Red: 7 Blue: 15 Yellow: 20Worst-Case Scenario
The worst-case scenario would involve the person taking all the non-yellow candies before taking any yellow candies. This means the person would first take all the green, red, and blue candies. The total count of these non-yellow candies is:
5 (green) 7 (red) 15 (blue) 27 non-yellow candies.
Once all 27 non-yellow candies have been taken, the next two candies taken must be yellow to ensure that there are at least two yellow candies. Therefore, the total number of candies that must be taken is:
27 (non-yellow) 2 (yellow) 29 candies.
Thus, the least number of candies he has to take to ensure that at least two of them are yellow is 29.
Understanding the Logic
When the person starts drawing candies, the worst-case scenario is that all the 5 green, 7 red, and 15 blue candies are drawn first, leaving only yellow candies. The next two candies drawn must be from the remaining 20 yellow candies, ensuring that at least two yellow candies are taken.
Mathematical Interpretation
Mathematically, if we consider the worst-case scenario, the first 27 draws could be all non-yellow candies, leaving the next two draws to be yellow candies. The calculation can be further broken down as follows:
The total number of non-yellow candies is:
5 (green) 7 (red) 15 (blue) 27 candies.
The remaining 20 candies are yellow. The person needs to take at least 2 more candies to ensure they get at least two yellow candies, as the first 27 draws could be all non-yellow.
Probability Implications
It's also worth considering the probability of drawing 27 candies from a box of 47 and not getting any yellows. The probability can be calculated as:
The total number of ways to choose 27 candies out of 47 is given by 47C27.
The number of ways to choose 27 candies from the 27 non-yellow candies is 27C27.
The probability of drawing 27 candies from the 47 and not getting any yellows is:
27C27 / 47C27 ≈ 1 in 9.76 trillion.
This underscores the rarity of such an event, further validating the worst-case scenario approach as a robust method for ensuring at least two yellow candies are drawn.
Conclusion
In conclusion, by taking a systematic approach and considering the worst-case scenario, we can ensure that at least two yellow candies are drawn from the box. The minimum number of candies needed to guarantee this outcome is 29. This problem not only tests logical reasoning but also provides insights into probability and the importance of considering all possible scenarios.