Question

A bowl contains raspberry and orange lollipops, with 15 of each. How many must be drawn one at a time to ensure that you have at least three orange lollipops?

Short answer

Draw 18 lollipops to guarantee at least three orange ones.

Step-by-step solution

Understanding the Requirement

To solve the problem, we need to determine the maximum number of lollipops one might draw without having at least three orange lollipops. The question involves ensuring that three orange lollipops are drawn among 30 total lollipops, made up of 15 raspberry and 15 orange lollipops.

Scenario Analysis

Consider the worst-case scenario where you draw the maximum number of raspberry lollipops before getting the desired orange ones. If you draw all 15 raspberry lollipops first, you still haven't drawn any orange lollipops.

Ensure Three Orange Lollipops

After drawing all 15 raspberry lollipops, you need to draw at least 3 more lollipops to ensure that these are orange. This is because there are only orange lollipops left to draw after all raspberries have been drawn.

Key concepts

Probability

In the context of combinatorics, probability helps us understand the likelihood of specific outcomes when drawing lollipops from the bowl. When calculating probabilities, we first identify the total number of possible results, which in this case is drawing any lollipop from a mix of 30 (15 raspberry and 15 orange). Probabilities are expressed as fractions with the number of favorable outcomes over the total outcomes. For instance, if asked for the probability of drawing one orange lollipop on the first try, the answer would be \( \frac{15}{30} = \frac{1}{2} \), since there are 15 orange lollipops and a total of 30.

• The probability doesn't directly solve the problem of ensuring three orange lollipops, but understanding it gives insight into potential outcomes.
• Knowledge of probability helps in making informed assumptions about the draws.

Even without calculating specific probabilities in this exercise, it's helpful to know how randomness might affect drawing sequences.

Worst-case scenario

Analyzing worst-case scenarios is crucial in problems where ensuring a specific outcome is required. In the lollipop selection problem, the worst-case scenario is when the goal of obtaining three orange lollipops is delayed to the maximum possible extent. This involves first drawing all the raspberry lollipops, which means drawing 15 raspberry lollipops before any orange one appears. This scenario assumes the slowest possible route to getting the needed outcomes.

• The purpose of considering the worst-case scenario is to plan for the situation where the desired outcome takes the longest time to achieve.
• In practicality, this is a safe way to ensure goals are met by preparing for the least favorable sequence of events.

So, it means drawing up to 18 lollipops to be sure that three of them are orange, ensuring no assumptions about likely sequences of drawing.

Lollipop selection problem

This problem is a classic example of a combinatorics exercise focused on ensuring certain probabilities are met within a structured selection process. The challenge is to decide how many lollipops must be picked to guarantee that at least three are orange, regardless of the actual randomness. Key points of solving this include:

• First, identify the total and desired outcomes, recognizing there are 15 of each type of lollipop.
• Then, consider how selecting without replacement changes the chances after each pick.
• Finally, ensure that by picking 18 lollipops in the worst-case scenario, you cover all possibilities where the last three could be orange.

This problem demonstrates foundational concepts in probability and planning under uncertainty, providing valuable lessons in strategy and structured thinking.