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Pre Calculus
9.7 Probability
Name _________________________
Date ____________________
Experiment Any occurrence for which the outcome is uncertain.
Sample Space The total number of possible outcomes for an experiment.
Event Any subset of the sample space.
Examples: a. What is the sample space for the rolling of two dice?
b. Having a total score of 9 is an event, how many outcomes are there for this event?
The probability of an event is a number derived from the ratio of n(E), the number of ways that a
favorable event can occur, and n(S), the number of outcomes in the sample space.
p(E) =
n( E )
n( S )
****all outcomes in the favorable event are equally likely to occur.
1
.
6
The number of ways a ‘2’ can be rolled is 1.The number of outcomes in the sample space is 6.
The probability of rolling a ‘2’ on a die is
If p(E) = 0, the event will never occur. If p(E) = 1, the event will absolutely occur. These are the two
extremes of probability. It means that 0  p(E)  1 .
Example: Consider the experiment where a bag contains perfectly made tiles labeled 1, 2, 3, 3, 4, 5, A, B,
B, C and D; and a tile is randomly selected from the bag.
a. What is the sample space?
How many members are in it?
b. What is the probability that a letter will be drawn?
c. What is the probability that a number will be drawn?
d. What is the probability that an even number will be drawn?
e. What is the probability that an F will be drawn?
f. What is the probability that either a number or a letter will be drawn?
The probability that an event will not occur is called the complement and is denoted p(E`).
p(E) + p(E`) = 1.
It follows that p(E) = 1 – p(E`) and p(E`) = 1 – p(E).
Example: If the probability of winning a poker game is
11
26
, the probability of losing the game is
.
37
37
Theoretical Probabilities
1. What is the probability of selecting a five card hand that is all spades from a deck of 52 cards?
b. What is the probability of selecting a five card hand that has no spades?
2. A bag contains 20 tennis balls, 4 of which are defective. If three balls are drawn what is the probability
that at least one is defective?
Try the following:
1. A town council is composed of 8 Democrats, 7 Republicans and 5 Independents. A committee of 3 is
chosen by selecting names from a hat. What is the probability that the committee has:
a. 2 Democrats and 1 Republican
b. 3 Independents
c. No Independents
d. 1 Democrat, 1 Republican, and 1 Independent
2. There are 18 golf balls in a bag, 6 are defective. If 4 are chosen at random:
a. What is the probability that all are defective?
b. What is the probability that at least one is defective?
3. There are 5 red marbles, 3 blue marbles and 2 white marbles in a bag. If 3 marbles are drawn at
random:
a. What is the probability that both white marbles are included?
b. What is the probability that there are no red marbles included?