Example

Example

Let suppose that a box contains 10 balls: 5 red, 2 blue, 2 green, and 1 yellow. The reference set is denoted $\Omega= \{r, b, g, y\}$ .

If we take one ball at random, the probability to get

a red one will be 0.5,
a blue one will be 0.2, a
green one will be 0.2, and
a yellow one 0.1.

We have at our disposal the whole possible knowledge about the events and thus, we can accurately quantify the chance of occurrence of each event.

Suppose now that somebody puts in another box: 4 red balls, 2 blue, 2 green, and 1 yellow, and then another ball whose colour is hidden. We can therefore split our mass of belief between the different events:

m(r)= 0.4,
m(b)= 0.2,
m(g)= 0.2,
m(y)= 0.1,
and the remaining part of 0.1, which we are unable to locate, and which we thus allot to $\Omega$ , as a whole.

We can then calculate the plausibility and the belief allotted to each event, as indicated on table 43.

The last ball being of unknown color, we can make 4 assumptions according to this color, each one leading to the determination of a probability distribution:

f_r if the ball is red,
f_b if the ball is blue,
f_g if the ball is green,
f_y if the ball is yellow.

It is checked that in each assumption, the probability of any event A lies between Bel(A) and Pl(A).

**Table 43:** Example of belief allotted to the occurrence of the colour of a ball.
Events	{r}	{b}	{g}	{y}	{r, b, g, y}	{r, b}...
Mass m	0.4	0.2	0.2	0.1	0.1	0
Bel	0.4	0.2	0.2	0.1	1	0.6
Pl	0.5	0.3	0.3	0.2	1	0.7
f_r	0.5	0.2	0.2	0.1	1	0.7
f_b	0.4	0.3	0.2	0.1	1	0.7
f_g	0.4	0.2	0.3	0.1	1	0.6
f_y	0.4	0.2	0.2	0.2	1	0.6

IRIT-UPS