Conditional probability (extended if required)

Imagine you have a big bag of marbles, some are red and some are blue. If you pick a marble without looking, the chance of getting a red one is simple probability. But what if I tell you that the marble you picked is definitely not blue? Now, the chances of it being red change, right? That's conditional probability! It's the chance of something happening, but only if something else has already happened.

Think of it like a detective solving a mystery. If the detective knows a certain suspect was at the crime scene (that's the 'condition'), it changes the probability (the 'chance') of them being the culprit. We're narrowing down our options based on new information.

We write it like this: P(A|B). This means 'the Probability of event A happening, given that event B has already happened.' The straight line '|' is like saying 'given that' or 'if we know that'.

Let's say you have a class of 30 students. 18 of them are girls, and 12 are boys. Now, out of all the students, 10 girls wear glasses, and 5 boys wear glasses.

Scenario 1: Simple Probability What's the probability that a randomly chosen student wears glasses? You'd count all students with glasses (10 girls + 5 boys = 15 students) and divide by the total students (30). So, P(Wears Glasses) = 15/30 = 1/2.

Scenario 2: Conditional Probability Now, here's the twist: What's the probability that a student wears glasses given that you already know the student is a girl? This is P(Wears Glasses | Is a Girl).

We only look at the girls now! There are 18 girls in total. Out of those 18 girls, 10 wear glasses. So, the probability becomes 10/18, which simplifies to 5/9. See how knowing the student is a girl changed the probability from 1/2 to 5/9? That's conditional probability in action!

1. Identify the Events: Figure out what event 'A' you're interested in (the one you want to find the probability of) and what event 'B' is the 'given' condition.
2. Find the Intersection: Calculate the probability of both events A and B happening together. This is written as P(A and B) or P(A ∩ B).
3. Find the Probability of the Condition: Calculate the probability of event B happening, P(B).
4. Divide: Use the formula: P(A|B) = P(A and B) / P(B). This means you divide the chance of both things happening by the chance of the condition happening.

Tree diagrams are super helpful for conditional probability, especially when events happen one after another. Imagine you're flipping a coin twice. The first flip has two branches (Heads or Tails). From each of those, the second flip also has two branches.

Each 'branch' of the tree shows a probability. When you go along a path (like Head, then Head), you multiply the probabilities on those branches to find the probability of that whole sequence happening. If you want to find the probability of an event given another, you look at the relevant branches and use the formula we just learned.

For example, if you want P(Second coin is Heads | First coin is Heads), you'd look at the branch where the first coin is Heads, and then only consider the probabilities coming off that branch for the second coin.

1. Confusing P(A|B) with P(B|A): ❌ Thinking P(A|B) is the same as P(B|A). These are often very different! Imagine P(cough | flu) (probability of cough if you have flu) vs. P(flu | cough) (probability of flu if you have a cough). They are not the same! ✅ Always carefully read which event is the 'given' condition. The event after the '|' is the condition.

2. Forgetting to Divide by P(B): ❌ Just finding P(A and B) and thinking that's the answer. ✅ Remember the formula: P(A|B) = P(A and B) / P(B). You must divide by the probability of the condition (P(B)) to 'zoom in' on only the cases where B happened.

3. Mixing Up Independent and Dependent Events: ❌ Assuming events are always independent (meaning one doesn't affect the other). If events A and B are independent, then P(A|B) is just P(A) because B doesn't change A's probability. ✅ Conditional probability is used when events are dependent (one event does affect the other). If P(A|B) is different from P(A), then the events are dependent.