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DEPENDENT EVENTS In
some cases one event is dependent on another; that is, two or more events are
said to be dependent if the occurrence
or nonoccurrence of one of the events affects the probabilities of occurrence
of any of the others. Consider
that two or more events are dependent. If p_{1} is the probability of a
first event; p_{2} the probability that after the first happens, the
second will occur; p_{3} the probability that after the first and
second have happened, the third will occur; etc., then the probability that all
events will happen in the given order is the product p_{1}  p_{2}
 p_{3}. EXAMPLE: A box contains 3 white marbles and 4 black marbles.
What is the probability of drawing 2 black marbles and 1 white marble in
succession without replacement? SOLUTION: On the first draw the probability of drawing a black marble is
On
the second draw the probability of drawing a black marble is
On the third draw the probability of drawing a white
marble is
Therefore, the probability of drawing 2 black marbles and
1 white marble is
EXAMPLE: Slips numbered 1 through 9 are placed in a box.
If 2 slips are drawn, without replacement, what is the probability that 1. both are odd? 2. both are even? SOLUTION: 1. The probability that the first is odd is
and the probability that the second is odd is
Therefore, the probability that both are odd is
2. The probability that the first is even is
and the probability that the second is even is
Therefore, the probability that both are even is
A second method of solution involves the use of
combinations. 1. A total of 9 slips are taken 2 at a time and 5 odd
slips are taken 2 at a time; therefore, _{
} 2. A total of _{9}C_{2 }choices and 4 even
slips are taken 2 at a time; therefore, _{
} PRACTICE PROBLEMS: In the following problems assume that no replacement is
made after each selection: 1. A box contains 5 white and 6 red marbles. What is the
probability of successfully drawing, in order, a red marble and then a white
marble? 2. A bag contains 3 red, 2 white, and 6 blue marbles. What
is the probability of drawing, in order, 2 red, 1 blue, and 2 white marbles? 3. Fifteen airmen are in the line crew. They must take
care of the coffee mess and line shack cleanup. They put slips numbered 1
through 15 in a hat and decide that anyone who draws a number divisible by 5
will be assigned the coffee mess and anyone who draws a number divisible by 4
will be assigned cleanup. The first person draws a 4, the second a 3, and the
third an 11. What is the probability that the fourth person to draw will be
assigned a. the coffee mess? b. the cleanup? ANSWERS:
