Least Likely Pick: What Balls Does Akmar Need?

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Least Likely Pick: What Balls Does Akmar Need?

Let's dive into a fun probability puzzle! Imagine Akmar has a bunch of balls in a container, and some of them have dots on them. Our mission is to figure out what kind of balls Akmar needs to add to this container so that if you were to randomly pick a ball, the dotted ball would be the least likely to be picked. He's got 10 identical balls to play with, so let's put on our thinking caps and get started.

Understanding the Problem

To make the dotted ball the least likely to be picked, Akmar needs to add balls that aren't dotted. Think of it like diluting a juice concentrate. The more water (non-dotted balls) you add, the less concentrated (less likely to pick a dotted ball) the juice becomes. Probability is all about ratios, guys, and we want to skew that ratio away from the dotted balls.

Here's the core idea: By adding more non-dotted balls, we increase the total number of balls in the container. This increase lowers the probability of picking a dotted ball because the dotted balls now represent a smaller fraction of the total.

Before we jump to conclusions, let's consider some extreme cases.

  • What if the container already has a million dotted balls? Adding 10 non-dotted balls won't make a huge difference, right? The dotted balls will still be overwhelmingly likely to be picked.
  • What if the container has only one dotted ball and one non-dotted ball? Adding 10 non-dotted balls will drastically reduce the likelihood of picking the dotted ball.

Key takeaway: The initial composition of the container matters. We need to add enough non-dotted balls to significantly shift the odds.

The Solution: Non-Dotted Balls

Akmar needs to add 10 non-dotted balls. This is the only way to decrease the probability of picking a dotted ball. Let's break down why:

  1. Increasing the Denominator: When you calculate probability, you're essentially creating a fraction: (Number of favorable outcomes) / (Total number of possible outcomes). In this case, the "favorable outcome" is picking a dotted ball. The "total number of possible outcomes" is the total number of balls in the container. Adding non-dotted balls increases the total number of balls (the denominator), which decreases the overall probability of picking a dotted ball.
  2. Maintaining the Numerator: Adding non-dotted balls doesn't change the number of dotted balls (the numerator). So, while the denominator goes up, the numerator stays the same, resulting in a smaller fraction (lower probability).

Example:

Let's say the container initially has 2 dotted balls and 3 non-dotted balls. The probability of picking a dotted ball is 2/5.

Now, Akmar adds 10 non-dotted balls. The container now has 2 dotted balls and 13 non-dotted balls. The probability of picking a dotted ball is now 2/15. See how the probability decreased?

Adding more dotted balls would do the opposite – it would increase the probability of picking a dotted ball, which is exactly what we don't want.

Why Other Types of Balls Won't Work

Let's quickly consider why adding other types of balls wouldn't achieve the desired result:

  • Dotted Balls: Adding more dotted balls would increase the likelihood of picking a dotted ball.
  • A Mix of Dotted and Non-Dotted Balls: This would depend on the ratio. If you add more non-dotted than dotted, it could decrease the probability, but the most effective and straightforward solution is to add only non-dotted balls.

Essentially, to minimize the chance of picking a dotted ball, you need to maximize the chance of picking a non-dotted ball. The only way to reliably do that with 10 identical balls is to make them all non-dotted.

A More Detailed Explanation with Scenarios

To really nail this down, let’s look at a few different starting scenarios and see how Akmar's additions affect the odds. We'll use the concept of probability, which, as we discussed, is calculated as:

Probability (Dotted Ball) = (Number of Dotted Balls) / (Total Number of Balls)

Scenario 1: Equal Quantities

  • Initial State: 5 Dotted Balls, 5 Non-Dotted Balls
  • Probability (Dotted Ball) = 5 / 10 = 0.5 (or 50%)
  • Akmar Adds 10 Non-Dotted Balls:
  • New State: 5 Dotted Balls, 15 Non-Dotted Balls
  • Probability (Dotted Ball) = 5 / 20 = 0.25 (or 25%)

Significant drop! The dotted ball is now much less likely to be picked.

Scenario 2: Dotted Ball Majority

  • Initial State: 8 Dotted Balls, 2 Non-Dotted Balls
  • Probability (Dotted Ball) = 8 / 10 = 0.8 (or 80%)
  • Akmar Adds 10 Non-Dotted Balls:
  • New State: 8 Dotted Balls, 12 Non-Dotted Balls
  • Probability (Dotted Ball) = 8 / 20 = 0.4 (or 40%)

Still a significant drop, even though dotted balls started with a big advantage. Adding the non-dotted balls made a big difference.

Scenario 3: Dotted Ball Minority

  • Initial State: 2 Dotted Balls, 8 Non-Dotted Balls
  • Probability (Dotted Ball) = 2 / 10 = 0.2 (or 20%)
  • Akmar Adds 10 Non-Dotted Balls:
  • New State: 2 Dotted Balls, 18 Non-Dotted Balls
  • Probability (Dotted Ball) = 2 / 20 = 0.1 (or 10%)

In this case, the dotted ball was already less likely to be picked, and Akmar's addition just made it even less likely.

What if Akmar Added Dotted Balls?

Let’s revisit Scenario 1 and see what happens if Akmar adds dotted balls instead.

  • Initial State: 5 Dotted Balls, 5 Non-Dotted Balls
  • Probability (Dotted Ball) = 5 / 10 = 0.5 (or 50%)
  • Akmar Adds 10 Dotted Balls:
  • New State: 15 Dotted Balls, 5 Non-Dotted Balls
  • Probability (Dotted Ball) = 15 / 20 = 0.75 (or 75%)

The probability increased dramatically! This is the opposite of what we wanted.

Conclusion: It's All About the Ratio!

So, to wrap it all up, Akmar needs to add 10 non-dotted balls to make the dotted ball the least likely to be picked. The key is to increase the proportion of non-dotted balls relative to dotted balls. By understanding the basic principles of probability and how ratios work, we can confidently solve this kind of puzzle. Keep thinking critically, and you'll be able to tackle even more complex problems! Remember, it's all about understanding how adding or subtracting elements changes the overall probability. And in this case, adding more of what you don't want to pick is the perfect strategy! This was a fun math problem, wasn't it guys? Hope you enjoyed it! Now go and conquer those probability challenges!