3: Condintional Probabilty and the Multiplication Rule

find the probability of the complement

"*At least One*" type probabilities mean that sometimes we need to find the probability than an event occurs at least once several independent trials. What is the easiest way to calculate these probabilities?

*sampling WITHOUT replacement*

"*_ _ _* means to leave the first item out when sampling the second."

*sampling WITH replacement*

"*_ _ _* means to replace the first item drawn before sampling the second."

*conditional probability*

"A probability that is computed with knowledge of additional information is called a *_ _*." - the probability that one event will occur GIVEN THAT another event has occurred.

*multiplication rule*

"The *_ _* finds "and" probabilities by rewriting the conditional probability definition." P(A and B) = P(A)P(B|A)

*Independent*

"Two events are *_* of each other if knowing that one event will occur (or has occurred) does not change the probability that the other event occurs. - so, this means that P(B|A) = P(B). Knowing A occurred did not change the probability of B occurring.

probabilities

"We will demonstrate independence using *_*."

P(New York *and* Chicago *and* Los Angeles*) (P(New York)xP(Chicago)xP(Los Angeles) = .... (0.235)(0.258)(0.26) = 0.0158)

Example: "According to recent figures form the U.S. Census Bureau, the percentage of people under the age of 18 was 23.5% in New York City, 25.8% in Chicago, and 26.0% in Los Angeles. If one person is selected from each city, what is the probability that all of them are under 18?" What formula do use to solve this using the *multiplication rule*?

5/15, 4/14, 5/15 x 4/14

Example: Suppose you have a batch of 15 memory chips and 5 are defective. 1) Select 1 chip at random: P(first chip is defective) = *_/_* 2) Set it aside and draw another. P(second chip is defective) = *_/_* 3)P(first two chips are drawn defective) = *_/_ x _/_*

P(A)P(B)

For any two independent events A and B, the *multiplication rule* is P(A and B) = *____* "When two events, A and B, are INDEPENDENT, then P(B|A) = P(B), because knowing that A occurred does not affect the probability that B occurs."

P(B|A) = P(A and B)/P(A)

How do you calculate conditional probability aka P(B|A)?

True

True or false: Is tossing a coin and rolling a number cube are INDEPENDENT events.

No events occur

What is the compliment of "At least one even occurs"?

P(B|A)

What notates...? - Probability of B given A (conditional probability) - find that probability that B occurs given that you already know that A occurred.

with

When sampling WITH or WITHOUT the replacement, the draws are *independent*.

without

When sampling WITH or WITHOUT the replacement, the draws are *not independent*.