Learn to find the probabilities of independent and dependent events'

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Learn to find the probabilities of independent
and dependent events.
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Vocabulary
independent events dependent events
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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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Additional Example 1 Classifying Events as
Independent or Dependent
Determine if the events are dependent or
independent. A. 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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Try This Example 1
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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Additional Example 2A Finding the Probability of
Independent Events
Three separate boxes each have one blue marble
and one green marble. One marble is chosen from
each box. A. What is the probability of choosing
a blue marble 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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Additional Example 2B Finding the Probability of
Independent Events
B. 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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Additional Example 2C Finding the Probability of
Independent Events
C. What is the probability of choosing at least
one blue marble?
Think P(at least one blue) P(not blue, not
blue, not blue) 1.
P(not blue, not blue, not blue)
0.125
Multiply.
Subtract from 1 to find the probability of
choosing at least one blue marble.
1 0.125 0.875
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Try This Example 2A
Two boxes each contain 4 marbles red, blue,
green, and black. One marble is chosen from each
box. A. What is the probability of choosing a
blue marble 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)
0.0625
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Try This Example 2B
Two boxes each contain 4 marbles red, blue,
green, and black. One marble is chosen from each
box. B. What is the probability of choosing a
blue marble and then a red marble?
Multiply.
P(blue, red)
0.0625
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Try This Example 2C
Two boxes each contain 4 marbles red, blue,
green, and black. One marble is chosen from each
box. C. What is the probability of choosing at
least one blue marble?
Think P(at least one blue) P(not blue, not
blue) 1.
P(not blue, not blue)
0.5625
Multiply.
Subtract from 1 to find the probability of
choosing at least one blue marble.
1 0.5625 0.4375
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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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Additional Example 3A Find the Probability of
Dependent Events
The letters in the word dependent are placed in a
box. A. If two letters are chosen at random, what
is the probability that they will both be
consonants?
P(first consonant)
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Additional Example 3A Continued
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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Additional Example 3B Find the Probability of
Dependent Events
B. 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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Additional Example 3B Continued
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)
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Try This Example 3A
The letters in the phrase I Love Math are placed
in a box. A. If two letters are chosen at random,
what is the probability that they will both be
consonants?
P(first consonant)
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Try This Example 3A Continued
If the first letter chosen was a consonant, now
there would be 4 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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Try This Example 3B
B. 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 Try This 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 3 vowels and 8 total letters left in the
box.
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Try This Example 3B Continued
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)
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Lesson Quiz
Determine if each event is dependent or
independent. 1. drawing a red ball from a bucket
and then drawing a green ball without replacing
the first 2. spinning a 7 on a spinner three
times in a row 3. A bucket contains 5 yellow and
7 red balls. If 2 balls are selected randomly
without replacement, what is the probability
that they will both be yellow?
dependent
independent