Topic 6 - Understanding Chance

Flashcards covering the fundamentals of probability theory, the Prosecutor's Fallacy, Binomial models, and simulation techniques in R.

What does chance or probability mean? 🤔

It's like predicting how often something will happen if you repeat the same thing many times. For example, if you flip a coin, you'll expect 50% heads over time!

What are the bounds for probability? 🏗️

All probabilities are between 0 and 1. Think of it like a scale: 0 is impossible and 1 is certain.

What is the complement rule in probability? 🥳

If you know the chance of something happening, just take 1 minus that to find the chance of it NOT happening! 🎉 For example: if $P(A)=0.7$, then $P(\text{not } A) = 1 - 0.7 = 0.3$!

What is the Prosecutor's Fallacy? 🧠

It's a mix-up where you confuse different probabilities! Like thinking the chance of finding evidence IF innocent is the same as being innocent IF evidence is found. They're not the same! ⚖️

What lesson does the Sydney Murder Case teach us? 🧩

A tiny chance of a DNA match doesn’t mean a small chance of being innocent. Think “TINY doesn’t equal TINY”! It's a tricky trap you might fall into.

What key errors happened in the Sally Clark Case? ⚠️

Three big mistakes: thinking of SIDS as independent events, mixing up with the Prosecutor's Fallacy, and ignoring natural causes! It shows why statistics matter!

What is conditional probability? 🎲

It’s like asking, what’s the chance of Event A happening now that I know Event B happened? Sort of like saying, **'What’s the chance you’ll find *chocolate* knowing it’s a sweet shop?'**

What does it mean for events to be independent? 🔗

One event's occurrence doesn’t change the other’s chance. Like flipping a coin doesn't change the weather! 🌤️ Remember: $P(B|A) = P(B)$.

What are dependent events? 📉

Here, the occurrence of one does affect the other! Imagine if rain makes you take an umbrella: one thing changes the next. $P(B|A) \neq P(B)$.

How do you use the Multiplication Rule? 🔄

To find the chance of two events happening together, you go: $P(A \text{ and } B) = P(A) \times P(B|A)$. If they’re independent, it’s just $P(A) \times P(B)$!

What’s the Addition Rule? 🍭

To find the chance of picking either of two candies, add the probabilities but remember to subtract the overlap (if any)! 😊 $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

What does it mean if two events are mutually exclusive? 🚫

They can’t happen at the same time! Like flipping a coin—you can’t have heads and tails! 🌟

What is a binary trial? 🔑

It’s a simple experiment with two possible outcomes! Think of it like a game where you either win or lose—just two choices!

What is the binomial model? 📊

It's a way to keep track of how many successes you get in several trials. You need a fixed number of tries and success/failure outcomes. Think of it like a basketball game deciding scores!

What’s the Binomial Formula? ⚖️

It's the math for finding a specific number of successes: $P(X = x) = C(n,x) \times p^x \times (1-p)^{n-x}$. Just remember count, success, and fails!

How do you use sample() in R? 📈

It’s like drawing names from a hat! You pick items from a bigger group, choosing how many and if you want repeats. Easy peasy!

What does replicate() do in R? 🔄

It’s like hitting repeat! It runs the same function several times and collects all the results for you.

What does set.seed() mean in R? 🎯

It's making random numbers predictable! Like saying, every time you play this game, you get the same random prize!🎁.

What does dbinom(x, n, p) calculate in R? 🏁

It finds the exact chance of getting exactly x successes in n trials! It’s like aiming for a specific score in a game!

What does pbinom(x, n, p) do? 🌠

Calculates the cumulative chance of getting x successes or fewer! Sort of like counting how many times you passed a test instead of just passing once!

How can you find the chance for 'at least x' successes in R? 🎉

Use 1 - pbinom(x-1, n, p)! It’s like saying, you want to know how often you get more than x wins, not just x! Remember: just flip it!