# Multiplication rule for "AND"

### Multiplication rule for "AND"

#### Lessons

$\cdot$ P(A and B): probability of event A occurring and then event B occurring in successive trials.

$\cdot$ P(B | A): probability of event B occurring, given that event A has already occurred.

$\cdot$ P(A and B) = P(A) $\cdot$ P(B | A)

$\cdot$ Independent Events
If the events A, B are independent, then the knowledge that event A has occurred has no effect on the probably of the event B occurring, that is P(B | A) = P(B).
As a result, for independent events: P(A and B) = P(A) $\cdot$ P(B | A)
= P(A) $\cdot$ P(B)
• Introduction
P(A and B) VS. P(A or B)

P(A and B): probability of event A occurring and then event B occurring in successive trials.
P(A or B):
probability of event A occurring or event B occurring during a single trial.

• 1.
Multiplication Rule for "AND"
A coin is tossed, and then a die is rolled.
What is the probability that the coin shows a head and the die shows a 4?

• 2.
Independent Events VS. Dependent Events
a)
One card is drawn from a standard deck of 52 cards and is not replaced. A second card is then drawn.
Consider the following events:
A = {the $1^{st}$ card is an ace}
B = {the $2^{nd}$ card is an ace}
Determine:
$\cdot$ P(A)
$\cdot$ P(B)
$\cdot$ Are events A, B dependent or independent?
$\cdot$ P(A and B), using both the tree diagram and formula

b)
One card is drawn from a standard deck of 52 cards and is replaced. A second card is then drawn.
Consider the following events:
A = {the $1^{st}$ card is an ace}
B = {the $2^{nd}$ card is an ace}
Determine:
$\cdot$ P(A)
$\cdot$ P(B)
$\cdot$ Are events A, B dependent or independent?
$\cdot$ P(A and B), using both the tree diagram and formula

• 3.
Bag A contains 2 red balls and 3 green balls. Bag B contains 1 red ball and 4 green balls.
A fair die is rolled: if a 1 or 2 comes up, a ball is randomly selected from Bag A;
if a 3, 4, 5, or 6 comes up, a ball is randomly selected from Bag B.
a)
What is the probability of selecting a green ball from Bag A?

b)
What is the probability of selecting a green ball?