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." Examine the spinner to the left. Are all of the outcomes equally likely? 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. B A C Examine the spinner. Are all of the outcomes equally likely? The theoretical possibility of spinning a B is: 1/3 (I out of 3)or 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 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.