Multiplication Rules

MULTIPLICATION RULES

We want to find the probability that event A occurs in a first trial and event B occurs in a second trial.

We associated the word “or” with the addition rules for probability, now we will associate the word “and” with the multiplication rules of probability.

The formula for multiplication rule of probability of independent events: P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial) P(A and B) = P(A) * P(B)

Example: Toss a coin and choose a primary color chip from a box.

What is the probability of getting a head on the coin and a red chip? Let A = the event of flipping the coin Let B = the event of choosing a color chip The sample space for A: heads and tails (two) The sample space for B: red, blue, and yellow (three) P(head and red) = P(head) * P(red) = 1/2 * 1/3 = 1/6 In this example the events were independent of each other.

Event A did not affect the occurrence of event B. One could have switched the order of the events with no change in the probability of the outcomes.

Should the occurrence of event A affect the probability of the occurrence of event B, the events are considered to be dependent. When events are dependent we still want to find the probability of event A occurring first followed by the occurrence of event B, but we must take into account what occurred in event A.

This is referred to as B/A or “B given A.” This leads to the multiplication rule of probability of dependent events: P(A and B) = P(A) * P(B/A)

Example: An urn contains five blue marbles and three red marbles. Two marbles are selected at random without replacement. Find the probability of selecting two red marbles. Let A = the event of choosing the first marble Let B = the event of choosing the second marble

The sample space for A: eight marbles The sample space for B: seven marbles P(red and red) = P(red) * P(red) = 3/8 * 2/7 = 3/28

Reference from MA120 Park University