Coins, Marbles and Dice
Key words: Experimental, theoretical, experiment, outcome
Objectives
• Estimate probabilities from experimental data.
• Understand that increasing the number of times and
experiment is repeated generally leads to better estimates of
probability.
• Compare experimental and theoretical probabilities in
different contexts.
Instructions for the teacher:
This is the second of two activities using an application called
Prob Sim. This, as the name suggests, is a probability
simulator. As before, you are strongly recommended to do the
experiments with real equipment before using the simulator.
KS3 Framework
page ref: 283, 285
Press APPS
Use the cursor ▼ or ▲ to find the Prob Sim application
Choose Prob Sim
Press ENTER
This is the main screen of the application.
The aim of this activity is to estimate the theoretical probability
from the experimental probability and to form a view of the
extent to which the experimental is a good indicator of the
theoretical. Note particularly that however many trials the
experimental probabilities of all outcomes will not be equal to
the theoretical probabilities – this would be very unlikely!
Choose 1:Toss Coins
Press WINDOW (for TOSS)
Make a table on the board:
Throw
Outcome
Fraction
Heads
Fraction
Tails
Repeat with single coin throws and fill in the table.
Throw
Outcome
Fraction
Fraction
Heads
Tails
1
H
1
0
1
1
2
H
2
0
2
2
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3
T
4
H
2
3
3
4
1
3
1
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Do this up to 10 single coin tosses.
Now increase the number of coin tosses:
Press ZOOM (for +10)
Fill in the table:
20
Bit less than
½
Bit more
than ½
Repeat a few times
Now increase the number of coin tosses:
Press TRACE (for +50)
Fill in the table.
Questions:
“What do you notice about the probabilities of getting a head or
a tail” – they get closer to ½ as we toss the coin more
“Are their ever exactly half heads or tails?” – No, the chance of
that would be very low.
Repeat this experiment with other types of trials, e.g. dice, pick
marbles and spinner.
Press Y= (for ESC)
Press Y= (for ESC again)
Press Y= (for YES)
This will return you to the main menu, where you can choose a
different simulation.
In each standard case the probabilities of each outcome are
equal. However, in the pick marbles simulation, it is possible to
modify the composition to make the outcomes different:
Choose 3.Pick marbles
Press ZOOM (for SET) and press WINDOW (for ADV)
You can now choose how many of each marble is in the bag. A
good starting setting would be 1 of marble A, 2 of marble B
and none of the others.
Press Y= (for ESC) twice. This takes you back to the
simulation.
When you look at the settling bar chart after a large number of
trials the column for marble B will be twice as high. This will
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require a discussion to be clear that the probabilities are and
3
2
.
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Coins, Marbles and Dice
First: Reset your TI 83-Plus
• Press 2nd + to get MEM
• Choose 7 (Reset…)
• Choose 1 (All Ram…)
• Choose 2 (Reset)
Use the calculator probability simulator to look at how probabilities in experiments
change.
1.
First set up your calculator:
Press APPS
Use the cursor ▼ or ▲ to find the Prob Sim application
Choose the letter next to (Prob Sim) –
it will certainly be different on your machine!
Press ENTER
Choose 1.Toss Coins
Press WINDOW (for TOSS)
Now you can press any of:
Press WINDOW (for +1)
Press ZOOM (for +10
Press TRACE (for +50)
2.
Make a copy of this table in your exercise book. (About 20 rows long):
Throw
3.
Outcome
Fraction
Heads
Fraction
Tails
Press +1 to throw the coin. Fill in the fraction of heads and tails you have got so
far. I rolled a head, so heads are 1 out of 1 and tails are 0 out of 1.
Throw
Outcome
1
H
Fraction
Heads
1
1
Fraction
Tails
0
1
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4.
Press +1 again. Fill in your table.
Keep going until you have filled in 10 coin throws.
5.
Press +10. Watch 10 more coin throws all in one go!
Fill in the table: remember this is now 20 coin throws in total.
Now you can only write an approximate fraction.
For example write either a bit more than ½, a bit less than ½ or exactly ½
6.
Press +10 again. Fill in your table. Do this 5 more times.
7.
Press +50. Watch 50 more coin throws all in one go!
Fill in the table.
Remember to add an extra 50 coin throws to the total.
8.
Repeat with +50 until your whole table is filled up.
9.
Answer these questions in your exercise book:
(a) What do you notice about the probabilities of getting a head or a
tail?
(b) Are there ever exactly half heads or tails?
11.
Press …
Y= (for ESC)
Y= (for ESC)
Y= (for YES)
… to return to the main Prob Sim menu.
(a) Repeat the experiment rolling a dice.
Use a table like this:
Toss
Outcome
Fraction of
1’s
1
3
0
1
(b)
Repeat the experiment spinning a spinner.
Use a table like this:
Toss
Outcome
Fraction of
1’s
1
3
0
1
Fraction of
6’s
0
1
Fraction of
4’s
0
1
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*Advanced*
12.
13.
Return to the main Prob Sim menu.
Choose 3 for pick marbles.
Repeat the experiment using a table like this:
Toss
Outcome
Fraction of
A’s
1
B
0
1
Return to the main Prob Sim menu.
Choose 3 for pick marbles.
Press ZOOM (for SET)
Press WINDOW (for ADV)
Type 1 to set one Marble A.
Use the down cursor ▼
Type 2 to set two Marble Bs.
Use the down arrow ▼ and type 0 to set all of the other marbles to zero.
Press Y= (for ESC)
Press Y= (for ESC again)
This takes you back to the simulation.
Now repeat the experiment with the same table as before.
(a)
(b)
(c)
(d)
14.
Fraction of
B’s
1
1
What is the probability of drawing a marble A (remember there are 3
marbles in the bag)?
What is the probability of drawing a marble B?
What do you notice about the probabilities of getting marble A or B?
Are they ever in the exact proportion?
Repeat the experiment with different settings for the numbers of marbles.
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