Probability Notes: Independent vs Dependent, Replacement, Sample Space, and Key Rules

Understanding the difference between independent and dependent events is key in probability. The main idea is whether the outcome of one event changes what might happen in the next event.

Independent Events

  • Definition: The outcome of one event does not affect the probability of other events.

  • Think of it like: Each event is a fresh start, unaffected by what came before.

  • Conditions: Often happens "with replacement" (meaning an item is put back after being chosen).

  • Examples:

    • Tossing a fair coin: If you toss a coin multiple times, getting heads on the first toss doesn't make it more or less likely to get heads on the second toss. Each toss has a P(\text{Heads}) = \frac{1}{2}.

    • Rolling a fair die: The chance of rolling a 6 is always \frac{1}{6}, no matter what you rolled before.

    • Drawing a card with replacement: If you pick a card, note it, and put it back in the deck (then shuffle), the probability of drawing any specific card remains the same for the next draw. The "sample space" (all possible outcomes) doesn't change.

Dependent Events

  • Definition: The outcome of one event changes the probabilities for the subsequent events.

  • Think of it like: What happens first influences what can happen next. The "world" changes after each event.

  • Conditions: Often happens "without replacement" (meaning an item is not put back after being chosen).

  • Examples:

    • Picking students from a class: If you pick one student from 30, then pick a second student, the second pick is now from 29 students. The probabilities change because the pool of choices gets smaller.

    • Drawing a card without replacement: If you pick a card and keep it, the deck now has one fewer card. The sample space shrinks (e.g., from 52 to 51 cards), so the probabilities for the next draw are different. If you drew an Ace, it's less likely to draw another Ace next.

Sample Space and Replacement

  • Sample Space (S): This is simply the list of all possible outcomes for an event.

  • With replacement: When you put an item back, the sample space stays the same size, and probabilities don't change. Draws are independent.

  • Without replacement: When you don't put an item back, the sample space gets smaller, and probabilities change. Draws are dependent.

Key Probability Rules

  1. Multiplication Rule (for finding the probability of a sequence of events):

    • For Independent Events (A and B):
      To find the probability that both A and B happen, you simply multiply their individual probabilities:
      P(A \cap B) = P(A) \cdot P(B)
      Example: Probability of getting Heads on two coin tosses is \frac{1}{2} \cdot \frac{1}{2} = \frac{1}{4}. (Like going along a path on a "tree diagram")

    • For Dependent Events (A and B):
      You need to use conditional probability.
      P(A \cap B) = P(A) \cdot P(B|A)
      Where P(B|A) means "the probability of B happening given that A has already happened."
      Example: Drawing two red balls without replacement. The probability of the second red ball depends on if the first one was red.

  2. Addition Rule (for "OR" events):

    • Mutually Exclusive (Disjoint) Events: These are events that cannot happen at the same time. Their intersection is empty (A \cap B = \emptyset).
      To find the probability that A or B happens, you just add their probabilities:
      P(A \cup B) = P(A) + P(B)
      \text{if } A \cap B = \emptyset)
      Example: Rolling a die and getting a 1 OR a 2. You can't get both at once, so P(1 \cup 2) = P(1) + P(2) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6}.

    • Non-Mutually Exclusive Events: If A and B can happen at the same time, you have to subtract the overlap to avoid double-counting:
      P(A \cup B) = P(A) + P(B) - P(A \cap B)

Practical Example (Red Ball Scenario)

  • Imagine a bag with 1 red ball out of 10 total (P(\text{red})=\frac{1}{10}).

  • If you add 80 more red balls, you now have 81 red balls out of 90 total.

  • The new probability of drawing a red ball is \frac{81}{90} = \frac{9}{10}.

  • Takeaway: Changing the composition of your sample space directly changes the probabilities of outcomes, even if it's still a single draw.