Theoretical probability is displayed
in the opening screen of the virtual coin below. The green line
shows the theoretical probability for heads and tails is 50%.
Experiment probability:
Toss a coin 20 times using the virtual coin provided by clicking
the "?" button. The experimental
probability does not have to match the theoretical probability.
Click the "Reset" button if you wish
to complete additional experimental probability trials.
The
theoretical probability of the event is the fraction:
# ways the event can occur
total possible outcomes
The
experimental probability for equally likely events is the
fraction:
# favourable outcomes
# trials
In the Experiment, the theoretical probability
of the event that heads comes up is:
P(H)
=
1/2
= 0.5 =
50%
P(T)
=
1/2
= 0.5 =
50%
In the Experiment,
if heads comes up 13 of 20 times, the experimental probability
of the event that heads comes up is:
Total possible outcomes are required for
theoretical probability calculations. A tree diagram (Method
1), table (Method 2) or list (Method
3) shows the possibilities. A tree diagram will
include the possibilities and the branches, while the chart
generally illustrates the same possibilities without the
branches.
Explore the total possible outcomes of the following pairs
of events.
Two Coins
Method
1
Complete a tree diagram to show
all of the possible outcomes of flipping two coins.
Compare it to the tree diagram below. It
should include branches connecting the 2
possibilities for Coin 1 with
the 2
possibilities for Coin 2. There
are 4 possible outcomes when you flip two coins.
Another way of drawing the ttree
diagram follows:
Event
A: Coin Flip
Event
B: Coin Flip
H
H
T
T
H
T
number of possible outcomes = 2
x 2
= 4
Method 2
Make a chart to show the 4 possibilities
for 2 coins. Compare your chart to the one given
below.
H
T
H
H, H
H, T
T
T, H
T, T
Method 3
Make a list of all the possibile
events (lists for 3 and 4 coins are included).
Compare your list to the one given below.
#Coins
#Possibilities
List
of possibilites
2
2
x 2 = 4
HH, HT,
TH, TT
Two
Six-sided Dice
Method 1
The following shows a tree diagram of the possible sums
when you toss two dice. To better understand the tree diagram,
roll Die 1 until you get 1. Find the corresponding
number in the chart below Die 1. Roll Die
2. It will display the value from 1 to 6.
The branches beside each event in the left cell shows
there are 6
possibilities for the second event. Each possibility for
Die 1 connects with the 6
possibilities for Die 2. There are
36 possibile outcomes when you roll two dice.
Event
A: Die Toss
Event
B: Die Toss
1
1
2
3
4
5
6
2
1
2
3
4
5
6
3
1
2
3
4
5
6
Event
A: Die Toss
Event
B: Die Toss
4
1
2
3
4
5
6
5
1
2
3
4
5
6
6
1
2
3
4
5
6
number of possible outcomes = 6
x 6 = 36
Method 2
Complete a chart to show the 36 possibilities
for 2 six-sided dice thrown. Compare your chart to the one
given below.
This is the standard probability table. Notice
that the possible outcomes of one die are placed across
the top of the chart and the possible outcomes for the second
die are placed down the left side of the chart. The sum
of the two die are placed in the appropriate table cell.
For example if the first and second die are both 1, then
the sum of the dice is 2.
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Method 3
Make a list of all the possibile events. Compare
your list to the one given below.
The following animation shows the pattern
of the possible sums when one tosses two. The 36 possible
combinations are also displayed. This information is used
in the next table to calculate the theoretical probability
for the sum of two dice tosses.
Theoretical probability for the sum
of two six-sided dice.
0
1/36
2/36
3/36
4/36
5/36
6/36
5/36
4/36
3/36
2/36
1/36
Six-sided Die and
Coin
Method
1
Complete a tree diagram to show
all of the possible outcomes of flipping a coin
and tossing a die. Compare your tree diagram
to the one given below. The table below shows
the 6
possibilities for Die Toss followed
by the 2 possibilities
for Coin Flip. There are 12 possible
outcomes when you roll a die and flip a coin.
Event
A: Die Toss
Event
B: Coin Flip
1
H
T
2
H
T
3
H
T
4
H
T
5
H
T
6
H
T
number of possible outcomes = 6
x 2
= 12
Method 2
Make a chart to show the 12 possibilities
for a coin and six-sided die tossed. Compare
your chart to the one given below.
H
T
1
1, H
1, T
2
2, H
2, T
3
3, H
3, T
4
4, H
4, T
5
5, H
5, T
6
6, H
6, T
Method 3
Make a list of all the possibile
events. Compare your list to the one given below.
#Dice
#Possibilities
List
of possibilites
2
6
x 2 = 12
1H, 1T
2H, 2T
3H, 3T
4H, 4T
5H, 5T
6H, 6T
5 Sector Spinner and Coin
Complete
a tree diagram to show all of the possible outcomes
of flipping a coin and spinning a 5 sector spinner.
Compare it to the tree diagram below. Branches
connect the 5Spinner possibilities with the
2
possibilities for Coin. There
are 10 possible outcomes when you spin a 5 sector
spinner and flip a coin.
Event
A: Spin Wheel
Event
B: Coin Flip
green
H
T
blue
H
T
white
H
T
grey
H
T
red
H
T
number of possible outcomes =
5
x 2
= 10
Remember when there are two independent events,
the number of possible outcomes equals the number
of possible outcomes of the first event times
the possible outcomes of the second event.
Independent Events: Probability of Tossing
Die and Spinning a Colour
Event
A: Toss Die
Event
B: Spin Wheel
Toss one six-sided dice
and spin a five sector color spinner.
What is the theoretical probability
of a particular number and color being
selected?
Solution:
You can use a real die
and spinner or use the virtual ones
above. The theoretical
probability of of any number
being tossed is 1/6, while the theoretical
probability of the spinner
stopping on each color is 1/5. Examples:
P(5, green)
means the probability of tossing
5 and spinning green
P(5,
green) = P(5)
x P(green) = 1/6
x 1/5 = 1/30 or 3.333...%
P(3, red) means
the probability of tossing
3 and spinning red
P(3, red)
= P(3) x P(red)
= 1/6 x 1/5 = 1/30 or 3.333...%
Complete 30 trials and complete
the experimental probability.
Two
Events
Theoretical
Probability
Experimental
Probability
P(5,
green) =
1/6
x 1/5 = 1/30 = 0.0333...3.3%
/30
P(3,
blue) =
1/6
x 1/5 = 1/30 = 0.0333...3.3%
/30
Independent
Events: Probability of Rolling Dice and Spinning
a Colour
Event
A: Toss Dice
Event
B: Spin Wheel
Toss two six-sided die and spin
a five sector color spinner. What is the
probability of a particular number and color being
selected?
Solution:
You can use real dice and spinner
or use the virt ual ones. The probability
of of any number being tossed varies, while the
probability of the spinner stopping on each color
is 1/5. Examples:
P(5,
green) means the probability of tossing
5 and spinning green
P(5,
green) = P(5) x
P(green) = 4/36 x 1/5 = 4/180
or 2.222...%
P(3, red)
means the probability of tossing 3
and spinning red
P(3,
red) = P(3) x P(red)
= 2/36 x 1/5 = 2/180 or 1.111...%
Complete 180 trials. Record the
number of times the selected two events occur
in the experiment, then complete the experimental
probability. Will the theoretical and
experimental probability always be the same? answer:
No
Two
Events
Theoretical
Probability
Experimental
Probability
P(5,
green) =
4/36
x 1/5 = 4/180 = 0.0222...2.2%
/180
P(3,
red) =
2/36
x 1/5 = 2/180 = 0.0111...1.1%
/180
Independent
Events: Probability of Drawing Two Cards
To be independent, you must put the first card back,
before drawing the second. This is equivalent
to drawing from two different decks of cards.
Complete the appropriate number
of trials needed for the experimental
probability. Will the theoretical and
experimental probability always be the same? answer:
No
Complete the appropriate number
of trials needed for the
experimental probability. Will
the theoretical and experimental probability always
be the same? answer:
No
Theoretical probability
is displayed in the opening screen of
each probability explorer selected. The green
line shows the theoretical probability for each
event.
Experiment probability:
Clicking the "?" button for Event
A and Event B starts
a probability experiment. The experimental
probability will likely approach the theoretical
probability for a large sample. Click
the "Reset" button
for Event A and Event
B if you wish to complete additional
experimental probability trials.
Note: the sectors for the
spinners included are the same size. Do not assume
that other resources will do the same. If you
are using a new spinner take a few moments to
examine the sector size.
3.3%