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The Probability of
Independent and
Dependent Events
8.12 The student will determine the
probability of independent and
dependent events with and without
replacement.
Prerequisite Skills
and Vocabulary
• Skills
• Making tree diagrams
• Understanding fractions, decimals,
and percents
• Vocabulary
• Equally likely events or outcomes
• Sample space
What do we mean by independent
and dependent events?
• Independent events - the occurrence of one
event does not affect the occurrence of the
other.
• Dependent events - the occurrence of one
event changes the probability that another
event will occur.
Independent Events:
What is the probability of flipping a coin twice
such that it comes up heads both times?
• The first flip landing on
heads or tails does not
affect whether or not
the second flip will land
on heads or tails.
• A tree diagram shows
each possible
outcome.
Since only one path gives two heads, the
probability of two heads is one out of four or 1/4.
Independent Events
• Drawbacks to using tree diagrams
• They can get very big very quickly.
• They can take a long time to create.
• We would like to be able to determine the
probability of two events without have to list
out all of the possible outcomes.
• Determine each probability and multiply them
together.
Independent Events:
What is the probability of rolling a three on a standard
number cube and flipping a coin and getting tails?
• These two events are
independent because the
rolling of the number cube
does not affect the flipping
of the coin.
• A tree diagram shows each
possible outcome.
Twelve possible outcomes, and
rolling a three and getting heads
is one option, so we have a
probability of 1/12.
Dependent Events
• If we have a jar with two red marbles and three blue
marbles, what is the probability that if we pick two
marbles we will pick one red and one blue from the
jar?
• If we do not put the first marble we pick back in the
jar, what we grab first from the jar will affect the
probability of our second pick.
• A tree diagram would be rather large for this problem
having 20 branches.
Dependent Events:
If we have a jar with 2 red marbles and 3 blue marbles, what
is the probability that we will pick one red and one blue?
Dependent Events:
If we have a jar with 2 red marbles and 3 blue marbles, what
is the probability that we will pick one red and one blue?
• Once we pick the first marble, there are only four left.
The size of our sample space has changed from 5
marbles to 4 marbles.
• The probability of picking a red marble first is 2/5.
• The probability of picking a blue marble second is
3/4.
• The product of these two probabilities is 6/20 or 3/10.
• What if we picked up the blue marble first and then
the red marble? Does it affect the probability?
Dependent Events
• If we draw two cards from a standard 52-card deck
without putting the cards back in the deck, what is
the probability they will both be jacks?
• The probability that one card is a jack is 4/52.
• Sample space size changes to 51 cards.
• The probability that the second card is a jack is 3/51.
• The product of these two probabilities is
1
 4  3   1  1 
 0.45%
         
52 51
13 17
221
Are the events dependent or
independent?
(That is does the size of the sample space change or does it stay the same)
1. Two cards are drawn from a standard deck of 52 cards.
(a) selecting two hearts when the first card is replaced (or put back)
in the deck.
(b) selecting two hearts when the first cards is not replaced (or put
back) in the deck.
(c) a queen is drawn, is not replaced, and then a king is drawn.
(d) drawing two reds cards (with replacement) from a deck of
cards.
(e) drawing a black card and then drawing a red seven (without
replacement) from a deck of cards.
Are the events dependent or
independent?
(That is, does the size of the sample space change or does it stay the same?)
2. A bag contains 9 Snickers, 7 Milky Way, and 4 Three
Musketeers candy bars. Determine whether the events are
dependent or independent and then find the probability.
(a) Drawing two Snickers, replacing the first candy bar in the
bag before drawing the second candy bar. (You don’t
want a Snickers bar so you are trying to get a different
kind.)
(b) Drawing Snickers, setting it aside, and then drawing a
Three Musketeers.
(c) Drawing a Milky Way, replacing it, and drawing a Three
Musketeers.
Are the events dependent or
independent?
(That is, does the size of the sample space change or does it stay the same?)
3. Rolling a four and then a five on a 12-sided number
cube.
4. Picking two green marbles out of a jar (without
replacement) given that there are 10 green marbles
and 8 blue marbles.
Discussion
• What did you learn from this session?
• How would you apply this to your classroom?
• What is still unclear?
• Comments and/or concerns?