Download File

Lesson 33
Finding the probability of
independent and dependent events
probability
• The probability of an event is the number of
favorable outcomes divided by the number
of total possible outcomes
• Number of favorable outcomes
• Number of possible outcomes
Independent events
• Events where the outcome of one does not affect the
probability of the other are called independent
events.
• To find the probability of 2 independent events,
multiply the probabilities of the 2 events.
• Flipping a coin is an independent event
• Spinning a spinner is an independent event
• Rolling a dice is an independent event
• The result of one does not affect the result of the
other
examples
• You have a coin and a spinner with 5 spaces:
• What is the probability of spinning a 3 and landing on
heads?
• These are independent so:
• P(3 and heads) = P(3) x P(heads)
•
=1
x
1 = 1
•
5
2
10
• What is the probability of spinning a 5 and then a 1?
• These are independent events so:
• P(5 and 1) = P(5) x P(1)
•
= 1 x 1 = 1
•
5
5
25
Dependent events
• With dependent events, the outcome of one
event does affect the probability of the other
event.
• Sometimes dependent events are described as
events without replacement.
• Example: drawing a red marble out of a bag and
not replacing it.
• The 2nd draw is affected by not putting it back in
the bag after you draw it out, because the number
of possible outcomes changes, and sometimes the
number of favorable outcomes also changes.
Independent or dependent?
• 1) rolling a 6 on a dice and rolling a 4 on another
dice
• 2) drawing a marble from a bag, putting it back in the
bag, and then drawing a blue marble
• 3) rolling a 6 on a dice and then rolling a 4 on the
same dice
• 4) drawing a red marble from a bag, not putting it
back, and then drawing a blue marble
• 5) tossing 2 coins, a dime and a quarter
• 6)putting 10 students' names in a bag and drawing 2
names without replacing the first name drawn
Using a tree diagram
• 1st
2nd
•
• H
•
H
HH
T
HT
•
• T
•
H
TH
T
TT
outcomes
Calculating the probability of
dependent events
• Joe has 2 squares and 3 circles in a bag.
• Find the probability of drawing a circle, keeping it, and
then drawing another circle.
• P(1st circle) = 3/5
• For the 2nd draw, a circle has been removed, so now there
are only 2 circles and only 4 shapes left
• So P(2nd circle)= 2/4
• So P(1st circle) x P(2nd circle) = 3 x 1 = 3
•
5
2
10
Find probability
• A bag has 5 $10 bills and 15 $1 bills:
• What is the probability of drawing a $1
bill, keeping it, and then drawing a $10
bill?
• What is the probability of drawing a $10
bill, keeping it, and then drawing another
$10 bill?
odds
• Odds are another way of describing
the likelihood of an event.
• Odds are expressed as ratios,
usually written with a colon.
• Odds can be calculated for
something or against something
happening.
Definition of odds
• Odds of an event: a ratio expressing the
likelihood of an event.
• Assume all outcomes are equally likely, and
there are m favorable and n unfavorable
outcomes.
• The sum of the favorable outcomes and the
unfavorable outcomes should be the same as
the total possible outcomes.
• The odds for the event are:
m:n
• The odds against the event are: n:m
Calculating odds
• A bag contains 6 red marbles, 2 yellow
marbles, and 1 blue marble.
• 1) What are the odds of drawing a red
marble?
• There are 6 favorable outcomes (6 red
marbles
• There are 3 unfavorable outcomes (3 not red)
• So odds are: 6:3 or 2:1
odds
• 1) What are the odds of drawing a blue
marble if there are 6 red, 2 yellow and 1
blue marbles.
• 2) What are the odds against drawing a blue
marble?
problem
• The computer club has 10 boys and 12 girls.
The chess club has 15 boys and 5 girls. Each
club will choose a member at random to be
club treasurer.
• 1) What is the probability that both treasurers
will be girls?
• 2) Both girls resign and a different pair of
students is chosen. What is the probability that
both treasurers will again be girls?