Understanding Probability: The Multiplication Rule and Beyond

What does it mean when two events are considered independent?

If the probability of raining on a day is $$\frac{1}{2}$$ and the probability of finding a lost item is $$\frac{1}{4}$$ , assuming these events are independent, what is the probability of both happening?

If the probability of choosing a red marble from a bag is $$\frac{1}{5}$$ and, after taking one out (and not replacing it), the probability of choosing a blue marble is $$\frac{1}{4}$$ , what is the probability of choosing a red marble and then a blue marble?

What is conditional probability?

What does it mean when two events are considered dependent?

Using the multiplication rule, how do you calculate the probability of two independent events, A and B, happening together?

If the probability of event A happening is $$\frac{3}{4}$$ and the conditional probability of event B happening given A is $$\frac{2}{3}$$ , what is the probability of both A and B happening?

What is the purpose of a tree diagram in probability?

If a tree diagram shows two branches for flipping a coin (heads or tails) and three branches for rolling a die (1, 2, or 3), how many possible outcomes are there for flipping the coin and then rolling the die?

In a deck of cards, if the probability of drawing a heart is $$\frac{1}{4}$$ and the probability of drawing an ace from the remaining cards (without replacing the first card) is $$\frac{1}{17}$$ , what is the probability of drawing a heart and then an ace?