To solve problems about events with different chances of happening, we need to understand some basic ideas about probability. Here’s a simpler breakdown:

1. Basic Concepts:

   • The probability of an event, like event A, is written as ( P(A) ). It can range from 0 to 1.
   • All possible outcomes together add up to 1. This means:
     ( P(A) + P(A') = 1 )
     Here, ( A' ) means everything that isn’t event A.

2. Rules of Probability:

   • For events A and B that don’t affect each other (independent events):
     ( P(A \cap B) = P(A) \times P(B) )
     This means we can multiply their probabilities to find the chance of both happening together.
   • For events where one affects the other (dependent events):
     ( P(A | B) = \frac{P(A \cap B)}{P(B)} )
     Here, the probability of A happening after B has occurred is found by dividing.

3. Conditional Probability:

   • Conditional probability looks at how likely an event is based on the happening of another event:
     ( P(A | B) = \frac{P(A \cap B)}{P(B)} )
     So, if we know B happens, we see how it changes the chance for A.

4. Examples:

   • If ( P(A) = 0.3 ) and ( P(B) = 0.5 ), and they're independent, then:
     ( P(A \cap B) = 0.3 \times 0.5 = 0.15 )
     This tells us the chance of both A and B happening together.
   • If the events depend on each other, and we know ( P(B) = 0.5 ) and ( P(A \cap B) = 0.2 ), then:
     ( P(A | B) = \frac{0.2}{0.5} = 0.4 )
     This tells us that if B happens, A has a 40% chance of happening.

In summary, understanding these basic rules of probability helps us figure out how likely events are to occur, whether they're independent or dependent on each other.