Independent and Dependent Probability

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Independent and Dependent Probability
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A compound event is made up of one or more
separate events. To find the probability of a
compound event, you need to know if the events
are independent or dependent.
Events are independent events if the occurrence
of one event does not affect the probability of
the other. Events are dependent events if the
occurrence of one does affect the probability of
the other.
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Example 1

• Determine if the events are dependent or
  independent.
• getting tails on a coin toss and rolling a 6 on a
  number cube
• B. getting 2 red gumballs out of a gumball
  machine

Tossing a coin does not affect rolling a number
cube, so the two events are independent.
After getting one red gumball out of a gumball
machine, the chances for getting the second red
gumball have changed, so the two events are
dependent.
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Example 2
Determine if the events are dependent or
independent A. rolling a 6 two times in a row
with the same number cube B. a computer
randomly generating two of the same numbers in a
row
The first roll of the number cube does not affect
the second roll, so the events are independent.
The first randomly generated number does not
affect the second randomly generated number, so
the two events are independent.
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Three separate boxes each have one blue marble
and one green marble. One marble is chosen from
each box. What is the P(blue) from each box?
The outcome of each choice does not affect the
outcome of the other choices, so the choices are
independent.
Multiply
P(blue, blue, blue)
0.125
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Independent
What is the probability of choosing a blue
marble, then a green marble, and then a blue
marble?
Multiply.
P(blue, green, blue)
0.125
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Dependent
To calculate the probability of two dependent
events occurring, do the following 1. Calculate
the probability of the first event. 2. Calculate
the probability that the second event would occur
if the first event had already occurred. 3.
Multiply the probabilities.
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Dependent
The letters in the word dependent are placed in a
box. If two letters are chosen at random, what is
the probability that they will both be consonants?
Because the first letter is not replaced, the
sample space is different for the second letter,
so the events are dependent. Find the probability
that the first letter chosen is a consonant.
P(first consonant)
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Dependent
If the first letter chosen was a consonant, now
there would be 5 consonants and a total of 8
letters left in the box. Find the probability
that the second letter chosen is a consonant.
P(second consonant)
Multiply.
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Dependent
If two letters are chosen at random, what is the
probability that they will both be consonants or
both be vowels?
There are two possibilities 2 consonants or 2
vowels. The probability of 2 consonants was
calculated in Example 3A. Now find the
probability of getting 2 vowels.
Find the probability that the first letter chosen
is a vowel.
P(first vowel)
If the first letter chosen was a vowel, there are
now only 2 vowels and 8 total letters left in the
box.
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Dependent
Find the probability that the second letter
chosen is a vowel.
P(second vowel)
Multiply.
The events of both consonants and both vowels are
mutually exclusive, so you can add their
probabilities.
P(consonant) P(vowel)