Theoretical and Estimated Probabilities

Theoretical probability is what we would expect to get as an outcome based on their probability. Like tossing a coin 10 times, we would get heads 5 times…in theory. But is that a guarantee of what we will get?

The probabilities formed using the sample space and what we know about coins are called the theoretical probabilities. Using observed relative frequencies is another method to estimate the probabilities of heads or tails. A relative frequency is the proportion derived from the number of the observed outcomes of an event divided by the total number of outcomes.

Here is an example of how to calculate relative frequency:We are flipping a coin 10 times and recording the outcomes.

Toss 1 2 3 4 5 6 7 8 9 10

Results H H T H H H T T T H

We now create a table like this……………….

    Total # of Relative  Total # of Relative 

Toss Outcome heads so far Frequency of  tails so far Frequency of 

      heads so far   tails so far

1 H 1 1/1 or 1 0 0/1 or 0

2 H 2 2/2 or 1 0 2/2 or 0

3 T 2 2/3 or .67 1 1/3 or .33

4 H 3 3/4 or .75 1 1/4 or .25

5 H 4 4/5 or .8 1 1/5 or .2

6 H 5 5/6 or .83 1 1/6 or .17

7 T 5 5/7 or .71 2 2/7 or .29

8 T 5 5/8 or .63 3 3/8 or .27

9 T 5 5/9 or .56 4 4/9 or .64

10 H 6 6/10 or .6 4 4/10 or .4

What is the sum of the relative frequency of heads and the relative frequency of tails for each row of the table?

    Total # of Relative  Total # of Relative 

Toss Outcome heads so far Frequency of  tails so far Frequency of 

      heads so far   tails so far

1 H 1 1/1 or 1 0 0/1 or 0

2 H 2 2/2 or 1 0 2/2 or 0

3 T 2 2/3 or .67 1 1/3 or .33

4 H 3 3/4 or .75 1 1/4 or .25

5 H 4 4/5 or .8 1 1/5 or .2

6 H 5 5/6 or .83 1 1/6 or .17

7 T 5 5/7 or .71 2 2/7 or .29

8 T 5 5/8 or .63 3 3/8 or .27

9 T 5 5/9 or .56 4 4/9 or .64

10 H 6 6/10 or .6 4 4/10 or .4

If you add their

frequencies

together, they

equal 1 or

100% !!

When we graph the results from HEADS, this is what the line graph would look like. Do you see a trend as more tosses are made?

The trend is heading down…getting closer to our theoretical probability.

Class Assignment:

Using the worksheet given to you, plot the tosses for heads on your graph and draw a conclusion about the TREND that you see. EXPLAIN YOUR REASONING for the trend. Also calculate the relative frequency.

Make sure you put your NAME on your paper and turn it in before you leave!!