INTRODUCTION TO LIKELY AND UNLIKELY

🎲 Relevance of the Theme: The theme "Likely and Unlikely" is a gem of mathematics that helps to understand the world of chances. Like a treasure hunter, by exploring what is likely and what is unlikely, we learn to predict the future. Not the distant future of the stars, but the next roll of a die or the next card that might come up in a deck. It's almost like having a mathematical crystal ball! This predictive ability is super useful, not only for games but for making everyday decisions.

🌍 Contextualization: In the vast ocean of mathematics, the study of "Likely and Unlikely" is like learning to navigate using the stars. It is part of Probability, a branch of mathematics that helps us understand the events that happen around us. Whether it's flipping coins, choosing cards, or deciding whether or not to take an umbrella, probability is there. In our curriculum, it is one of the first steps to understanding how to quantify the chances of events. This prepares us for deeper seas in mathematics and critical thinking, which we will explore in the coming years.

THEORETICAL DEVELOPMENT

Components

• Chance: It's like a measure of the possibilities of something happening. Think of a bag of candies: if we have more strawberry candies than lemon, the chance of picking a strawberry is higher.
• Event: Anything that can happen, like drawing a card from a deck. An event can be simple like "drawing a queen" or compound like "drawing a queen or a king".
• Outcome: It's what happens after you perform an action, like rolling a die. If you roll a die and it lands on 4, that's the outcome.

Key Terms

• Probability: The chance of an event happening, usually expressed in percentage. For example, the probability of flipping a head when tossing a coin is 50%.
• Likely Event: An event that has a high chance of happening. Like rolling an even number on a standard die, which has three even numbers (2, 4, 6) and three odd numbers (1, 3, 5), the chance of even is 50%.
• Unlikely Event: An event that has a small chance of happening. Like rolling the number 6 on a six-sided die, there is only one chance in six, so it is unlikely.

Examples and Cases

• Dice Roll: When rolling a standard six-sided die, each number (1 to 6) has the same probability of coming up, which is approximately 16.67% (1 in 6). If we ask what the probability of rolling a number greater than 4 is, then we add up the chances of numbers 5 and 6, which gives us approximately 33.33%.
• Coin Toss: When we toss a coin, we have two possible outcomes: heads or tails. Both have the same chance of occurring, therefore, it is a likely event that will result in 50% for each side.
• Choosing Cards: In a deck of 52 cards, if we want to find an ace, there are 4 aces in the deck. The probability is 4 in 52, approximately 7.69%. This is much less likely than drawing any card of hearts, which are 13 in total - a chance of 25%.

DETAILED SUMMARY

Relevant Points:

• Measuring Chances: Understand that we can measure the chances of something happening and that this helps us predict outcomes.
• Equality of Conditions: Recognize that in some events, like tossing a coin, all outcomes have the same chance of occurring.
• Influence of the Number of Possibilities: Understand that the more options there are, the lower the chance of a specific event happening.
• Compound Events: Know that the probability of a compound event (like rolling an even number on a die) is the sum of the probabilities of each simple event that composes it.

Conclusions:

• Probability in Action: We identify that probability is a practical part of mathematics that is present in many everyday situations.
• More and Less Likely Outcomes: We learn to differentiate between likely and unlikely events and how this can influence our expectations and decisions.
• Use of Percentages: We conclude that probability is often expressed in percentage, which facilitates understanding and comparison between the chances of different events.

Exercises:

1. Dice Roll: If you roll a die twice, what is the probability of getting an odd number both times?
2. Magic Coin: Imagine you have a magic coin with heads on one side and tails on the other, but heads appear 70% of the time. If you toss this coin three times, what is the probability of getting tails at least once?
3. Deck Time: You have a deck of 52 cards and draw a card, then put it back and shuffle again. What is the probability of drawing a king in the first two attempts?