I. Notion of probabilities
1. Outcomes and probability tree
- An experiment is random when we cannot predict in advance what the outcome will be among the different possible outcomes.
- The diagram that allows us to visualize the different possible outcomes of a random experiment is called theprobability tree.
- Each branch of this probability tree indicates the probability of an outcome.
- We say that the tree is weighted by the probabilities.
Katia throws a balanced die with six sides numbered 1,2,2,3,3 and 3.
We observe the number indicated on the upper face: the outcomes are 1, 2 and 3.
The die is balanced, so each side has as much chance of coming up as any other.
- Thus, the probability of the number 1 coming out is , since only one side of the die bears the number 1.
- Two faces have the number 2, so the probability of outcome 2 is or .
- Similarly, that of outcome 3 is or or another 0.5 or 50%.
These results are summarized in the probability tree below:
- A probability is a number between 0 and 1.
- The probability is expressed as a number in fractional or decimal form, or as a percentage.
2. the events
- An event realized by no outcome is called an impossible event and its probability is 0.
- An event realized by all the outcomes is called a certain event and its probability is 1.
- The opposite event to an event A, noted , is realized when A is not.
The probability of an event is the sum of the probabilities of the outcomes that realize it.
the sum of the probabilities of an event and its opposite is equal to 1.
Two events are incompatible when they cannot occur at the same time.
For Katia’s die, the event “get an even number” and the event “get 3” are incompatible.
3. from frequencies to probabilities
When no consideration of regularity or symmetry allows us to know the probability of an outcome, we can estimate it by performing a random experiment a large number of times .
Let be a random experiment and an event A whose probability is noted P(A).
When this random experiment is repeated a very large number of times , the frequency of occurrence of event A tends to stabilize around P(A).
By throwing a large number of times a cork in the same way, we could estimate
the probability that it will fall back into one of the following positions:
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