Solved Problems: Analyzing Differences in Probability for Grade 7 Math

📚 Understanding Differences in Probability

Probability is the chance of something happening. Sometimes, two situations might seem similar, but the probability of each outcome can be quite different. Let's explore this with some examples.

🎲 Definition of Independent Events

Independent events are events where the outcome of one event does not affect the outcome of the other. For example, flipping a coin and rolling a die are independent events.

• 🪙 Example: Flipping a coin twice. The result of the first flip doesn't change the possible results of the second flip.
• ➕ Formula: If events A and B are independent, then $P(A \text{ and } B) = P(A) \times P(B)$

🎯 Definition of Dependent Events

Dependent events are events where the outcome of one event does affect the outcome of the other. For example, drawing two cards from a deck without replacing the first card is a dependent event.

• 🃏 Example: Drawing two cards from a deck. The probability of drawing a specific card changes after the first card is drawn (and not replaced).
• ➗ Formula: If events A and B are dependent, then $P(A \text{ and } B) = P(A) \times P(B|A)$, where $P(B|A)$ is the probability of B given that A has occurred.

📊 Comparison Table: Independent vs. Dependent Events

🔑 Key Takeaways

• 💡 Identify the Relationship: Determine if one event influences the other. This is the core of distinguishing between independent and dependent events.
• ✍️ Apply the Correct Formula: Use the appropriate formula based on whether the events are independent or dependent.
• 🤔 Consider Replacement: In drawing scenarios, consider whether items are replaced. If they are not replaced, it's likely a dependent event.