In A Bag Of Lollipops, There Are 10 Red, 7 Orange, 5 Purple, And 8 Green Lollipops. If Two Lollipops Are Randomly Drawn One After The Other Without Replacement, What Is The Probability That Both Lollipops Are Red?

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In this comprehensive guide, we will delve into the fascinating world of probability, tackling a classic problem involving lollipops in a bag. We'll break down the problem step-by-step, ensuring a clear understanding of the concepts and calculations involved. Our main goal is to determine the probability of drawing two red lollipops consecutively from a bag containing a mix of different colored lollipops, without replacing the first one drawn. This problem perfectly illustrates the concept of dependent events in probability, where the outcome of the first event affects the probability of the subsequent event.

Understanding the Problem

Let's first clearly define the scenario. Imagine a bag filled with an assortment of lollipops: 10 red, 7 orange, 5 purple, and 8 green. We are interested in the probability of a specific outcome: drawing two red lollipops in a row, without putting the first lollipop back into the bag. This "without replacement" condition is crucial because it means that the total number of lollipops, and the number of red lollipops, changes after the first draw. This change directly influences the probability of drawing a red lollipop on the second draw. This scenario highlights the concept of conditional probability, where the probability of an event depends on the occurrence of a previous event.

To effectively tackle this problem, we need to understand some key probability concepts:

  • Probability: The likelihood of an event occurring, expressed as a ratio of favorable outcomes to total possible outcomes.
  • Independent Events: Events where the outcome of one does not affect the outcome of the other.
  • Dependent Events: Events where the outcome of one event does affect the outcome of the other.
  • Conditional Probability: The probability of an event occurring given that another event has already occurred.

In our lollipop problem, the two draws are dependent events. The act of removing a lollipop on the first draw changes the composition of the bag, thus influencing the probability of the second draw. We will use the principles of conditional probability to accurately calculate the desired outcome.

Step 1: Calculate the Probability of Drawing a Red Lollipop on the First Draw

To begin, we need to determine the probability of drawing a red lollipop on the very first try. This is a straightforward application of the basic probability formula:

Probability (Event) = (Number of favorable outcomes) / (Total number of possible outcomes)

In this case:

  • Favorable Outcomes: Drawing a red lollipop. There are 10 red lollipops.
  • Total Possible Outcomes: Drawing any lollipop. There are 10 (red) + 7 (orange) + 5 (purple) + 8 (green) = 30 lollipops in total.

Therefore, the probability of drawing a red lollipop on the first draw is:

Probability (Red on First Draw) = 10 / 30 = 1 / 3

This means that there is a 1 in 3 chance of picking a red lollipop on the first try. This initial probability forms the foundation for our subsequent calculations. We've established the likelihood of the first event, and now we need to consider how this event impacts the probability of the second.

Step 2: Calculate the Probability of Drawing a Second Red Lollipop, Given That One Red Lollipop Was Already Drawn

This is where the concept of conditional probability comes into play. We've already drawn one red lollipop and not replaced it. This significantly changes the landscape of the problem. Now, the bag contains:

  • 9 red lollipops (since we removed one)
  • A total of 29 lollipops (since we removed one)

So, the probability of drawing another red lollipop on the second draw, given that we drew a red lollipop on the first draw, is:

Probability (Red on Second Draw | Red on First Draw) = 9 / 29

This notation, "P(Red on Second Draw | Red on First Draw)", is the standard way to represent conditional probability. It reads as "the probability of drawing a red lollipop on the second draw given that a red lollipop was drawn on the first draw." The probability has changed because the total number of lollipops and the number of red lollipops have both decreased.

This step is crucial because it highlights the dependence between the two events. The probability of the second event is directly influenced by the outcome of the first event. If we had replaced the first lollipop, the probability of the second draw would have remained 10/30, and the events would have been independent.

Step 3: Calculate the Overall Probability of Drawing Two Red Lollipops in a Row

To find the probability of both events happening in sequence, we need to multiply the individual probabilities. This is a fundamental rule of probability: the probability of two dependent events occurring is the product of the probability of the first event and the conditional probability of the second event given that the first event has occurred.

Probability (Red on First Draw AND Red on Second Draw) = Probability (Red on First Draw) * Probability (Red on Second Draw | Red on First Draw)

Plugging in the values we calculated earlier:

Probability (Two Red Lollipops) = (1 / 3) * (9 / 29) = 9 / 87

We can simplify this fraction by dividing both the numerator and denominator by 3:

Probability (Two Red Lollipops) = 3 / 29

Therefore, the probability of drawing two red lollipops in a row from the bag, without replacement, is 3/29. This is our final answer, representing the likelihood of the specific outcome we set out to determine. This result underscores the importance of considering the impact of prior events when calculating probabilities, especially in situations involving dependent events.

Expressing the Probability as a Percentage

While the fraction 3/29 accurately represents the probability, it can be helpful to express it as a percentage for a more intuitive understanding. To do this, we simply divide the numerator by the denominator and multiply by 100:

(3 / 29) * 100 ≈ 10.34%

Therefore, there is approximately a 10.34% chance of drawing two red lollipops in a row from the bag, without replacement. This percentage provides a clear sense of the likelihood of this outcome. While not a highly probable event, it is certainly within the realm of possibility.

Key Takeaways and Applications

This lollipop problem provides a clear and practical illustration of key probability concepts, including dependent events and conditional probability. Understanding these concepts is crucial for solving a wide range of real-world problems, from risk assessment in finance to predicting outcomes in scientific experiments. Here are some key takeaways:

  • Dependent Events: When the outcome of one event affects the probability of another, the events are dependent. Failing to recognize dependence can lead to inaccurate probability calculations.
  • Conditional Probability: The probability of an event occurring given that another event has already occurred. This is a powerful tool for analyzing sequential events.
  • "Without Replacement" Matters: In problems involving drawing objects from a set, the "without replacement" condition significantly impacts the probabilities. Always carefully consider whether objects are being replaced or not.
  • Probability Multiplication Rule: To find the probability of multiple events occurring in sequence, multiply their individual probabilities (taking into account any dependencies).

The principles illustrated in this example have broad applications in various fields:

  • Games of Chance: Calculating probabilities in card games, lotteries, and other games of chance.
  • Medical Research: Assessing the effectiveness of treatments and predicting patient outcomes.
  • Finance: Evaluating investment risks and predicting market trends.
  • Quality Control: Determining the probability of defects in manufacturing processes.

By mastering these fundamental probability concepts, you can gain a deeper understanding of the world around you and make more informed decisions.

Conclusion

By carefully dissecting the lollipop problem, we've successfully calculated the probability of drawing two red lollipops in a row, without replacement. We've explored the concepts of dependent events, conditional probability, and the importance of accounting for changes in the sample space. This example serves as a valuable illustration of how probability principles can be applied to solve real-world problems. Remember to always carefully analyze the problem, identify dependencies between events, and apply the appropriate formulas to arrive at an accurate solution. Understanding probability is a crucial skill that empowers us to make informed decisions in a world filled with uncertainty.