Example 1 – tossing a coin 200 times
We tossed a coin 200 times, and then wrote down the results (”heads” or “tails”) for every toss.
We summarized the results in the following table:
The Value |
The Frequency (number of times that each value was received) |
The Relative Frequency (actual) |
The Probability (the expected relative frequency) |
“Heads” |
96 |
48% |
50% |
“Tails” |
104 |
52% |
50% |
Total |
200 |
100% |
100% |
Note that we divide the results into two groups (”heads” and “tails”), and examine the relative frequency of each group, which is then compared with the probability that we had calculated in advance for each group. As expected, the relative frequency is close to the theoretical probability, but not identical to it.
Example 2 – tossing a coin 1,000 times
In this example, we will toss the coin 1,000 times.
The results we received are displayed in the following table:
The Value |
The Frequency (number of times that each value was received) |
The Relative Frequency (actual) |
The Probability (the expected relative frequency) |
“Heads” |
510 |
51% |
50% |
“Tails” |
490 |
49% |
50% |
Total |
1,000 |
100% |
100% |
The relative frequency is closer to the theoretical probability than it was in the case of 200 tosses, but the results are still not identical.
Example 3 – tossing a coin 10,000 times
In this example, we will toss the coin a very large number of times (i.e., 10,000).
The results we received are displayed in the following table:
If we look at all three of these examples together, we see that in none of them is the relative frequency identical to the theoretical probability (i.e., 50% “heads”, 50% “tails”). At the same time, as the number of tosses increases, the relative frequency approaches the theoretical frequency.
Actually, the probability reflects the relative frequency that we would expect to obtain if we were to toss the coin an infinite number of times.
When throwing the dice, it is of course possible to receive any of the following six results: 1, 2, 3, 4, 5, 6.
The probability of each such result is 1/6, or 16.6%. As we saw when using a coin, we will present two examples with the dice, and we will again see that as the number of throws increases, the relative frequency obtained approaches the theoretical probability.
Example 1 – throwing the dice 120 times
We obtained the following results:
The value |
The Frequency (number of times that each value was received) |
The Relative Frequency (actual) |
The Probability (the expected relative frequency) |
1 |
15 |
12.5% |
16.6% |
2 |
22 |
18.3% |
16.6% |
3 |
26 |
21.6% |
16.6% |
4 |
21 |
17.5% |
16.6% |
5 |
10 |
8.3% |
16.6% |
6 |
26 |
21.6% |
16.6% |
Total |
120 |
100% |
100% |
Example 2 – throwing the dice 12,000 times
We obtained the following results:
The value |
The Frequency (number of times that each value was received) |
The Relative Frequency (actual) |
The Probability (the expected relative frequency) |
1 |
1,950 |
16.3% |
16.6% |
2 |
1,901 |
15.8% |
16.6% |
3 |
2,233 |
18.6% |
16.6% |
4 |
1,942 |
16.2% |
16.6% |
5 |
2,185 |
18.2% |
16.6% |
6 |
1,789 |
14.9% |
16.6% |
Total |
12,000 |
100% |
100% |
In both of these examples with the dice, the relative frequency of each value is not identical to the theoretical probability, but as the number of throws increases then the relative frequency approaches the theoretical probability.
Probabilities that Cannot Be Calculated in Advance
In the example of the coin and the dice, it is possible to predict in advance the probability of each group.
On the other hand, if we sort first grade children in the USA into 100 different height groups, we will not be able to calculate in advance the probability of each group.
In cases of this type, we can only estimate the probability on the basis of the results of the sample. We will discuss this subject later in the chapter.