STAT 203 Chapter 15 Probability Rules
• If all the possible outcomes in a sample space are equally likely, the probability of an
event A is given by:
P (A) =
number of outcomes in event A
total number of outcomes in the sample space
• conditional probability of B given A (A and B are events)
The notation P (B|A) denotes the conditional probability of event B given that event
A has occurred.
P (B|A) =
P (A and B)
P (A)
If A and B are independent events,
P (B|A) = P (B)
• If events A and B are disjoint, then the probability that event A or event B or both
will occur is
P (A or B) = P (A) + P (B)
[Addition Rule]
If events A and B are not disjoint,
P (A or B) = P (A) + P (B) − P (A and B)
[General Addition Rule]
• If events A and B are independent, then the probability that event A and event B will
occur together is
P (A and B) = P (A) × P (B)
[Multiplication Rule]
If events A and B are dependent,
P (A and B) 6= P (A) × P (B),
and one would have to obtain P (A and B) by
P (A and B) = P (A) × P (B|A)
or P (A and B) = P (B) × P (A|B)
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[General Multiplication Rule]
Examples
1. An urn contains 5 red marbles (R), 4 yellow marbles (Y) and 3 green marbles (G).
(a) You are to randomly draw one marble from the urn. What is the probability that
a red or a green urn is drawn?
(b) Now you draw two marbles at random without replacement. Given that you have
drawn a yellow marble in the first draw, what is the probability of drawing a green
in the second draw?
(c) Without any information on the result of the first draw as given in part (b), what
is the probability of drawing
i. two green marbles?
ii. one green and one red?
2. An ice cream stand reports that 12% of the cones they sell are “jumbo” size. A little
girl wants to see what a “jumbo” cone looks like, so she stands and watches the sales
for a while.
(a) What is the probability that the first jumbo cone is the fourth cone she sees them
sell?
(b) What is the probability that out of the first four cones sold, at least one is a
jumbo cone?
3. A survey of local car dealers revealed that 74% of all cars sold last month had CD
players, 68% had alarm systems, and 60% had both CD players and alarm systems.
(a) Are the event that the car has a CD player and the event that the car has an
alarm system disjoint? Explain.
(b) Are the event that the car has a CD player and the event that the car has an
alarm system independent? Explain.
(c) What is the probability that a randomly selected car had neither a CD player nor
an alarm system?
(d) What is the probability that a car had a CD player unprotected by an alarm
system?
(e) What is the probability a car with an alarm system had a CD player?
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