Transcript Math SCO G1 and G2

Gl:
Students will be expected to conduct
simple experiments to determine
probabilities
G2 Students will be expected to
determine simple theoretical
probabilities and use fractions to
describe them.
What is experimental probability?
 Experimental probabilities are calculated by
performing experiments.
 For example, if a die is rolled 60 times, the number
4 might come up 15 times giving an experimental
probability of 15/60.
 If a spinner was spun 30 times and the number 6
came up 4 times, then the experimental probability
of spinning a 6 would be 4/30.
 I rolled a pair of dice 25 times and the sum of the
numbers was 8 on 4 of the rolls. What is the
experimental probability that the sum is 8? Does
this seem reasonable?
Partner Work
 Roll a die 4 times. Create two 2-digit numbers and
subtract them. Record the difference between each
set of numbers in a chart. Repeat the experiment 20
times. Calculate the experimental probability that the
difference you get is less than 10. Then repeat the
experiment another 20 times and compare the
probability for 20 rolls compared to 40 rolls.
 Use spinner marked in 10 equal sections. Put the
point of your pencil and a paper clip in the center of
the spinner. Spin the clip five times and total the
numbers spun (find the sum of the 5 numbers you
spun). Repeat this 10 times. What is the
experimental probability that the sum of the five
numbers is greater than 25? Compare your findings
with others. If time permits, combine all the groups’
results for a class value.
What is Theoretical Probability?
 A Reminder: Experimental probabilities are
calculated by performing experiments. If a die is
rolled 20 times and the number 3 comes up 4 times,
the experimental probability of rolling a 3 is 4/20 (1
out of 5, or 20 percent or 0.20)
 Theoretical probabilities are calculated by
determining all the possible outcomes of an event
and then comparing how many times a particular
outcome (possibility) occurs to the total outcomes
(possibilities). If a die is rolled 60 times, the number
4 might come up 15 times. The theoretical
probability is 1/6 because there are six equally likely
possible outcomes (1, 2, 3, 4, 5, 6) when a die is
rolled and one of these outcomes is the number 4. 1
is compared to 6 to get the ratio 1/6. Theoretical
probability is what would happen in theory; this is
not always what really happens.
Finding Theoretical Probability
 Determine how many times a particular outcome
(possibility) exists in that situation. For example, in a coin
toss, rolling a head is one outcome. This becomes the
numerator of the fraction. The numerator of your theoretical
probability will be 1.
 Now look at the total possible outcomes you could get. This
becomes the denominator of your theoretical probability.
For example, when flipping a coin, there are two possible
outcomes of this event. You can flip “heads” or “tails”. There
are two outcomes in total. (Heads and Tails). So the
denominator of your theoretical probability will be 2.
 So the theoretical probability of getting “Heads” is 1 over 2
or ½ (also 50 % or 0.5). The same is true of flipping “Tails”.
 Sometimes when we perform an experiment enough times,
we can end up with the experimental probability being
almost exactly what the theoretical probability is.
Finding Theoretical Probability:
Another Example
 How do you calculate theoretical probability of tossing a 4
using a regular die?
 Determine how many times a particular outcome
(possibility) exists in that situation.
 The number 4 is found once on the die. So the numerator
needed for the theoretical probability is 1 since there is
only 1 four on the die.
 The total number of possible outcomes when you roll a die
is six. You could get a 1, 2, 3, 4, 5, or a 6 when you roll.
This means there are six possible outcomes. This becomes
the denominator of your theoretical probability. So the
denominator when finding the theoretical probability of
rolling a four is 6.
 So the theoretical probability of rolling a 4 is 1/6 (1 over 6).
The same is true of rolling any of the possible outcomes in
this situation.
In the case of the rolling of a die, all six numbers
have an equal chance of being rolled—we say that
all outcomes are "equally likely."
2
1
Examine the
spinner to the left.
Are all of the
outcomes equally
likely?
3
Even though there are three outcomes, they are not equally
likely. The theoretical probability of spinning a 1 is 1/2, not
1/3 (one out of three). This can be determined by
calculating the fractional part of the spinner covered by 1.
Describe the theoretical probability of spinning a B in
the situation shown below. Use both fractions and
decimals to give your answer.
A
B
Examine the
spinner. Are all of
the outcomes
equally likely?
C
The theoretical possibility of spinning a B is:
1/3 (I out of 3)
or 0.333 repeating
Describe the theoretical probability of rolling a 5 on a
regular dice. Use both fractions and decimals to give
your answer.
Examine the die.
Are all of the
outcomes equally
likely?
The theoretical possibility of rolling of a 5 is:
1/6 (I chance out of 6
possible outcomes)
or 0.666 repeating
Now roll a die 60 times and record the number of
times you rolled a 5. The experimental probability of
rolling a 5 can be calculated by creating a fraction
where the numerator is the number of times you
actually rolled a 5 during your experiment compared
to the number of rolls you made (60). How does this
compare to the theoretical probability of rolling a 5
which was 1/6 (1 out of 6)?
Describe the theoretical probability of rolling an even
number on a regular dice. Use both fractions and
decimals to give your answer.
Examine the die.
Are all of the
outcomes equally
likely?
The theoretical possibility of rolling of an even number is:
3/6 or 1/2 (I chance out
of 2 possible outcomes)
or 0.5
Student Activities
 Pretend that you are putting coloured cubes into a
bag. Draw coloured cubes so that the theoretical
probability of choosing a red one is 1/2 and
choosing a green one is 1/4. Why is there more
than one way to model this situation?
 List the first 20 multiples of 3 and determine the
probability that a multiple of 3 is also a multiple of
6 and is also a multiple of 9.
 Explain how to determine the theoretical
probability of rolling a 3 on a regular die. Next,
explain how this would change if the die contained
the numbers 1, 3, 3, 3, 5, and 6.
Extra Activities
Using a Hundred Chart to conduct
experiments
 Begin at a designated number and roll a die to determine
where to go next:
 1— down 1 and right 1
 2— down 2 and right 2
 3— down 3 and left 1
 4— down 4 and left 2
 5— down 5
 6— up 1
Determine the probability that after 5 rolls you will land in some
designated range of numbers (between … and …), or on a
certain type of number such as an even number or a multiple of
3.