Download D6 Probability

Probability
unlikely
possible
probable
50-50
chance
likely
poor
chance
certain
Independent Events
Vocabulary
A sample space is the set of all possible outcomes.
A simple example is flipping a coin.
The sample space is {heads, tails}
Vocabulary
Two events are independent if the outcome of one has no
effect on the outcome of the other.
Examples are rolling two dice, or
spinning a spinner and rolling a dice
Vocabulary
Two events are dependent if the outcome of one event
relies on the other event.
Examples are picking two marbles out of
a bag, or two socks out of a drawer
Vocabulary
The complement of an event is all the outcomes NOT
included in the event. It is shown by A’.
The following spinner is spun once:
What is the probability of it not landing on the yellow sector?
3
P(not yellow) OR P(yellow’) =
4
Vocabulary
The intersection of two events is all the outcomes that are
SHARED by both events. It is denoted by A∩B and can be
read A and B
Event A is even numbers on a dice
Event B is multiples of 3 on a dice
A ∩B is the number 6
P(A ∩B) = 1/6
If two events are independent, then their intersection can
be calculated as P(A ∩B)=P(A)*P(B)
Vocabulary
The union of two events is all the outcomes of either event
events. It is denoted by AUB and can be read A or B.
Event A is odd numbers on a dice
Event B is multiples of 3 on a dice
A UB is {1,3,5,6}
P(A UB) = 4/6 or 2/3
Union can be calculated as P(AUB)=P(A)+P(B)- P(A∩B)
A or B  A  B
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© Boardworks Ltd 2005
Example
• The two spinners are spun. What is the
probability that both spinners will show an even
number?
P(1st spinner even) = 4
8
P(2nd spinner even) = 4
8
P(both spinners even) =4 ∙ 4 = 1
8 8 4
Example
• A game uses a dice and a spinner.
1. A player rolls a dice. What is the P(odd
#)?
2. The player spins the spinner. What is the
P(red)?
3. What is the P(odd # and Red)?
Example
• There are 4 red, 8 yellow and 6 blue socks in a
drawer. Once a sock is selected it is not
replaced. Find the probability that 2 blue socks
are chosen.
P(1st blue sock) =
6
# of socks after 1 blue is
18
removed
P(2nd blue sock) =
5
17
Total # of socks after 1 blue is
removed
P(Two blue socks) =
6 ∙ 5=
18 17
5
51
Example
• In a certain town, the probability that a
person plays sports is 65%. The
probability that a person is between the
ages of 12 and 18 is 40%. The probability
that a person plays sports and is between
the ages of 12 and 18 is 25%. Are the
events independent?
Mutually exclusive outcomes
Outcomes are mutually exclusive if
they cannot happen at the same time.
For example, when you toss a single coin either it will land on
heads or it will land on tails. There are two mutually exclusive
outcomes.
Outcome A: Head
Outcome B: Tail
Mutually exclusive outcomes
A pupil is chosen at random from the class. Which of the
following pairs of outcomes are mutually exclusive?
Outcome A: the pupil has brown eyes.
Outcome B: the pupil has blue eyes.
These outcomes are mutually exclusive because a pupil can
either have brown eyes, blue eyes or another colour of eyes.
Outcome C: the pupil has black hair.
Outcome D: the pupil has wears glasses.
These outcomes are not mutually exclusive because a pupil
could have both black hair and wear glasses.
Adding mutually exclusive outcomes
If two outcomes are mutually exclusive then their probabilities
can be added together to find their combined probability.
For example, a game is played with the following cards:
What is the probability that a card is a moon or a sun?
1
1
and
P(sun) =
3
3
Drawing a moon and drawing a sun are mutually exclusive
outcomes so,
1
1
2
P(moon or sun) = P(moon) + P(sun) =
+
=
3
3
3
P(moon) =
Adding mutually exclusive outcomes
If two outcomes are mutually exclusive then their probabilities
can be added together to find their combined probability.
For example, a game is played with the following cards:
What is the probability that a card is yellow or a star?
1
1
P(yellow card) =
and
P(star) =
3
3
Drawing a yellow card and drawing a star are not mutually
exclusive outcomes because a card could be yellow and a star.
P (Y U S) = P(Y) + P(S) – P(Y∩S) = 1/3 + 1/3 – 1/9 = 5/9.
Venn Diagrams
• Venn diagrams are useful in figuring out
probabilities
P(AUB)’
Example
• If A is the students who own bikes and B is the
students who own skateboards, find
• A∩B and P(A ∩B). Are the events independent?
• AUB and P(AUB)
• (AUB)’ and P(AUB)’
Ex. Amongst a group of 20 students, 7 are taking Math
and of these 3 are also taking Biology. 5 are taking
neither. What is the probability that a student chosen at
random is taking Biology?
Solution: The diagram shows the 20 students.
20
The “eggs” show Math
and Biology
3 do both
5 do neither
B
M
4
3
5
7 take Math ( but we have 3 already )
Amongst a group of 20 students, 7 are taking Maths and
of these 3 are also taking Biology. 5 are taking neither.
What is the probability that a student chosen at random
is taking Biology?
Solution: The diagram shows the 20 students.
20
The final number (taking
Biology but not Math ) is
given by
20  4  3  5  8
B
M
4
3
3  8 11

So, P( student takes Biology ) =
20
20
8
5
Filling in a Venn diagram
100 people are asked if they eat meat, fish, both, or neither. You
are told that 55 eat meat, 52 eat fish, and 21 eat neither. Use
this information to complete the Venn diagram below.
21 eat neither
27
28
Now use the formula:
24
P(A È B)= P(A)+ P(B)- P(A Ç B)
79=55+52- P(A Ç B)
21
P(A Ç B)=107 -79
P(A Ç B)=28
Meat only=55-28
Fish only =52-28
Meat only=27
Fish only =24
21+24+28+27=100
Finding probabilities
Use the Venn diagram to find
the probability that someone
picked at random:
a) eats meat,
b) eats fish,
c) eats neither,
d) eats only fish,
e) eats both.
55
100
52
100
21
100
24
100
28
100
‘Given that’
Given that a man eats meat,
find the probability that he
also eats fish.
28
27 + 28
28
=
55
Summary of methods
The “or” rule
The word “or” often indicates that the probabilities need
to be added together.
P(A or B) = P(AUB) = P(A) + P(B) – P(A∩B)
The “and” rule
The word “and” often indicates that the probabilities need
to be multiplied together.
P(A and B) = P(A∩B) = P(A) × P(B) if the two events are
independent.