Independence and Conditional Probabilities as Probabilities

Algebra inside the conditioned world

Independence

Working definitions and a classical application

P(A) = 0.5, P(B) = 0.4, and P(A and B) = 0.3. Are A and B independent?

Two fair coins are tossed. Given that exactly one shows heads, what is the status of the events first coin heads and second coin heads?

Cards are drawn one by one without replacement from a shuffled deck. What is the probability the first two cards are both aces? Write it as a fraction in lowest terms and enter the denominator.

Two fair coins are tossed; A is first coin heads, B is second coin heads, C is the coins match. Compute P(A and B and C). Independence of all three would give 1/8; the true value as a fraction has what numerator?

A machine is up with probability 0.9. Two conditionally independent sensors each report correctly with probability 0.95. Both read UP. Rounded to two decimals, what is the probability the machine is up? Enter like 1.00.

In your own words, state the most important definition, theorem or method of this lesson and explain what it says.