Mathematics lesson on probability topic

1. Probability

2. Vocabulary
Probabilit
y
describes how likely
it is that some
event will happen

3. For example:
What is the
probability of
randomly
picking the king
of hearts from
a deck of
cards?

4. Vocabulary
Sample space:
a combination of
all of the possible
choices

5. For example:
When drawing a card from
a deck of 52 cards, there
are 52 possible choices
The sample space is all
of the 52 choices.

6. Sample space for
drawing a card from a
deck

7. Properties of
Probabilities
The probability of any
event must fall
between 0 and 1.
 0 < probability < 1

8. Properties of
probabilities
 If something is impossible,
then the probability is 0.
 The probability that
Thanksgiving will fall on a
Monday is 0.
 Thanksgiving never falls on
Monday.

9. Properties of
probabilities
 If an event is certain, then
the probability is 1.
 The probability that
Thanksgiving will fall on a
Thursday is 1.
 Thanksgiving always falls on
Thursday.

10. Which of the
following are possible
probabilities?
 0.2
 3.1
 3%
 -2.3%
 134%
 1/3
 3/2

11. Calculating Probabilities
 A pair of dice are rolled.
 What is the probability of rolling a seven?
 To calculate:
 Number of ways a seven could be rolled
Number of ways two dice can land

12. A pair of dice are rolled.
What is the probability of rolling a
seven? 6/36 which reduces to 1/6

13. Situation:
 In a local survey,
high school
students were
asked what kind
of music they
preferred. The
results are
printed.
Response Frequency
Country 656
Rap 202
Top 100 610
Jazz 48
Classic
Rock
1051

14. Problem:
 What is the
probability that a
randomly
selected student
will prefer Jazz?
Response Frequency
Country 656
Rap 202
Top 100 610
Jazz 48
Classic
Rock
1051
48 prefer Jazz
Total: 2567
Probability = 48/2567

15. Rap Concert:
 Cain has 3 tickets to a concert. Ulysses,
Armando, Aaron, and Jaime would all like to
go. Cain will randomly select two of them.
 What is the probability that Aaron and Jaime
will get to go?

16. Rap concert:
 What are the possible combinations?
1. Jaime and Armando
2. Jaime and Aaron
3. Jaime and Ulysses
4. Armando and Aaron
5. Armando and Ulysses
6. Ulysses and Aaron
Probability that
Jaime and
Aaron will get to
go = 1/6

17. Rap Concert:
 Cain has 3 tickets to a concert. Ulysses,
Armando, Aaron, and Jaime would all like to
go. Cain will randomly select two of them.
 What is the probability that Aaron or Jaime
will get to go?

18. Rap concert:
 What are the possible combinations?
 Jaime and Armando
 Jaime and Aaron
 Jaime and Ulysses
 Armando and Aaron
 Armando and Ulysses
 Ulysses and Aaron
Probability that
Jaime or Aaron
will get to go =
5/6

19.  Two members from a five member committee
are to be randomly selected to serve as
chairman and secretary.
 First person selected will be chairman.
Second person selected will be secretary.
 Five members are: Hope, Sara, Luis,
Elizabeth, and Ariel.
 What is the probability that Hope will be the
chairman and Sara will be the secretary?
Make charts to simplify:

20. Hope
Sara
Luis
Ariel
Elizabeth
Sara
Sara
Sara
Sara
Hope
Hope
Hope
Hope
Luis
Luis
Luis
Luis
Ariel
Ariel
Ariel
Ariel
Elizabeth
Elizabeth
Elizabeth
Elizabeth
Hope and Sara
Hope and Luis Hope and Elizabeth
Hope and Ariel
Sara and Hope
Sara and Luis Sara and Elizabeth
Sara and Ariel
Luis and Sara
Luis and Hope Luis and Elizabeth
Luis and Ariel
Ariel and Sara
Ariel and Hope Ariel and Elizabeth
Ariel and Luis
Elizabeth and Sara
Elizabeth and Hope Elizabeth and Ariel
Elizabeth and Luis
Probability
= 1/20

21. Tossing three coins:
 When three coins are tossed, the probability
of getting at least one tail is:
 Start by finding the possible combinations:
HHH TTT
HTT THH
HTH THT
HHT TTH
How many
possibilities include
at least one tail?
How many
possibilities total?
Probability:
7/8
7
8

22. Probabilities: OR
 What is the probability of drawing a king or a
heart from a deck of 52 cards?

23. OR means add (but do not
count the same item twice)
 Probability of selecting a heart: 13/52
 Probability of selecting a king: 4/52
 Probability of selecting a heart or a king:
 13/52 + 4/52 – 1/52 (because we can’t count
the king of hearts twice) = 16/52

24. Probabilities: OR
 What is the probability of drawing a king or a
heart from a deck of 52 cards?

25. AND means multiply:
unrelated events
 When a coin is tossed and then a die is
rolled, the probability of getting a tail on the
coin and an odd number on the die is:
 If the two events do not effect one another then
find the probability of each separately and multiply

26. Unrelated events
Tails Odd
Probability of getting tails: 1/2
Probability of odd number: 3/6

27. Unrelated events:
Tail and Odd
 Probability of getting tails: 1/2
 Probability of getting odd number: 3/6
 Probability of getting tails AND odd number:
=(1/2)(3/6)
= 3/12
= 1/4

28. What is the probability of drawing
a king AND then drawing a king
again if the first card is replaced?
Probability of drawing the first king: 4/52
Probability of drawing the second king: 4/52
Answer: (4/52)(4/52) = 1/169
Related or Unrelated?
Unrelated

29. AND means multiply:
related events
 What is the probability of drawing a king, not
replacing it, and drawing a king again?:
 If one event effects the outcome of the other
event then find the probability of the first event,
and then find the probability of the second event
remembering the effect of the first

30. What is the probability of drawing
a king AND then drawing a king
again if the first card is not
replaced?
Probability of drawing the first king: 4/52
Probability of drawing the second king: 3/51 (3
kings and 51 cards left)
Answer: (4/52)(3/51) = 1/221
Related or Unrelated?
Related

31. Sample Problems

32. Two dice are rolled. What is
the probability of getting
doubles or a sum of 10?

33. Probability of getting doubles
or sum of 10:
 Probability of getting doubles: 6/36
 Probability of getting sum of 10: 3/36
 Number of rolls that are both doubles and
sum of 10: (1 roll, the five and five roll)
 Probability answer:
 = 6/36 + 3/36 - 1/36
 = 8/36
 = 2/9

34.  A bag contains a red bead, a green bead,
and a blue bead. If a bead is selected
and its color noted, and then it is
replaced and another bead is selected,
the probability that both beads will be of
the same color is
Probability = 3/9 which reduces to 1/3
RG GR BR
RB GB BG
RR GG BB

35.  A box contains a penny, a nickel, a dime, and a
quarter. If a coin is selected and then replaced
and a 2nd coin is selected, the probability of
getting an amount greater than 11 cents is
Probability = 10/16 which reduces to 5/8

36.  A box contains a penny, a nickel, a dime,
and a quarter. If two coins are selected
without replacement, the probability of
getting an amount greater than 11 cents is
Probability = 8/12 which reduces to 2/3

37.  The probability that a family visits New
York is 0.64, and the probability that a
family rides on the Subway is 0.50. The
probability that a family does both is 0.40.
Find the probability that a family visits New
York or rides the subway.
NY
0.64
SUBWAY
0.50
BOTH
0.40
OR means add: probability of people who only went to NY
+ probability of people who only went to subway

38. NY
0.64
SUBWAY
0.50
BOTH
0.40
OR means add: probability of people who only went to NY
+ probability of people who only went to subway
Remember: Some of the people who went to the NY also went
to the subway, we don’t want to count those people twice.
Therefore we can subtract the people who went to both.
Probability of going to NY OR subway = 0.64 + 0.50 – 0.40
= 0.74

39.  If a pair of tetrahedral die are rolled, what
is the probability that the sum will be 6?
How many sides on a
tetrahedral dice?
Probability = 3/16
1
2
3
2
2
2
2
4
1
2
3
3
3
3
3
4
1
2
3
1
1
1
1
4
1
2
3
4
4
4
4
4
4 sides

40.  If a pair of tetrahedral die are rolled,
given that the sum is even, what is the
probability that the sum will be 6?
Probability = 3/8
1
2
3
2
2
2
2
4
1
2
3
3
3
3
3
4
1
2
3
1
1
1
1
4
1
2
3
4
4
4
4
4
4 sides