Probability
Probability
questions are becoming increasingly common. Probability questions
tend to be bundled among the difficult questions, so high scorers
will commonly encounter them. If you are a low scorer and are
pressed for time, consider skipping most of the material past
"Simple Probability." This is a computer-adaptive test,
and low scorers aren't likely to encounter the most difficult
probability question types.
A. Simple Probability
B. Probability of Multiple Events
C. Independent and Dependent Events
D. Mutually Exclusive Events
E. Conditional Probabilities
F. Combinations
A. Simple Probability
In general, the probability of
an event is the number of favorable outcomes divided by the total
number of possible outcomes.
Probability=
(# of favorable outcomes) / (# of possible outcomes)
Example 1
What
is the probability that a card drawn at random from a deck of
cards will be an ace?
Solution
In this case there are four favorable outcomes:
(1) the ace of spades
(2) the ace of hearts
(3) the ace of diamonds
(4) the ace of clubs.
Since each of
the 52 cards in the deck represents a possible outcome,
there are 52 possible outcomes. Therefore, the probability is
4/52 or 1/13.
The same principle
can be applied to the problem of determining the probability
of obtaining different totals from a pair of dice.
Example 2
What
is the probability that when a pair of six-sided dice are thrown,
the sum of the numbers equals 5?
Solution
There are 36 possible outcomes when a pair of dice is thrown.
Consider that if one of the dice rolled is a 1, there are six
possibilities for the other die. If one of the dice rolled a
2, the same is still true. And the same is true if one of the
dice is a 3,4,5, or 6. If this is still confusing, look at the
following (abbreviated) list of outcomes: [(1,1),(1,2),(1,3),(1,4),(1,5),(1,6);
(2,1),(2,2),(2,3)… (3,1),(3,2),3,3)… (4,1)…(5,1)…(6,1)….
The total number of outcomes is 6
× 6
= 36. Since four of the outcomes have a total of 5 [(1,4),(4,1),(2,3),(3,2)],
the probability of the two dice adding up to 5 is 4/36 = 1/9.
Example 3
What
is the probability that when a pair of six-sided dice are thrown,
the sum of the number equals 12?
Solution
We already know the total number of possible outcomes is 36,
and since there is only one outcome that sums to 12, (6,6--you
need to roll double sixes), the probability is simply 1/36.
NOTE: The material from here on through the
end of the section is dense and intended only for medium to high
scorers. Because this is a CAT (computer-adaptive test), it is
relatively unlikely that lower scorers will encounter these questions,
and, if they are short of time, they are better off putting their
time into other sections.
w B. Probability of Multiple Events
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