Are mutually exclusive events independent of the probability
The probability P (A) for a random event A gives you a measure of the certainty that it will occur. It is specified in the interval [0; 1]; The greater P (A), the more certain you can expect A to occur. A certain event has a probability of 1, while an impossible event has a probability of 0.
With the symmetrical dice, for example, each number has the same probability of being thrown, namely . If you consider the event of throwing a number smaller than 5, this can be achieved with numbers 1 to 4; its probability is then .
Your sure event, which has a probability of 1, would be to get a number less than 7; an impossible event with a probability of 0 of about 8.
How do you formally define probabilities?
Formally, the probability measure P is an assignment of the result space , so the possible events to of a random experiment, on real numbers in the interval [0; 1],
for which applies:
The probability of every possible event is between zero and one.
The probability that any of the possible events will occur is one.
The probability of the union of mutually exclusive events is equal to the sum of the probabilities of these events.
In the case of the symmetrical cube, the first two properties have already been shown above.
The probability that the union of all mutually exclusive events occurs is the sum of the individual probabilities:
Calculation rules for probabilities
The following calculation rules for probabilities can be derived from these properties, which you can easily illustrate with the help of Venn diagrams:
If is the counter-event to A, and P (A) is the probability with which the event A occurs, then the probability with which A does not occur is exactly (1-P (A))
When the meteorologists report that the probability of rain for tomorrow you also know the probability that it will stay dry; namely it amounts to .
Probability of two mutually exclusive events
If A and B are two mutually exclusive events (disjoint events), you can calculate the probability that their union will occur as the sum of the individual probabilities.
The numbers when rolling the dice are disjoint events, you can either roll a 1 or a 2, but not both together. Therefore, the following applies to the probability of rolling a number (AZ) less than 3:
Probability of any two events
If A and B are not mutually exclusive, but can also occur together, you calculate the probability of their union as the sum of the individual probabilities minus the probability of their intersection in order to avoid double counting.
You know that the mood of one of your fellow students depends on both the good weather and the results of their soccer team. You designate the event with A: “the sun is shining”, that with a probability occurs, and with B the event: "His soccer team wins", with a probability of . Both events can also occur together, as the intersection of A and B: : "The sun is shining and the soccer team wins", with a probability of .
The probability that either the sun is shining or the soccer team wins, i.e. the union of A and B, is then given as:
So you can with a probability of assume that your fellow student is in a reasonably good mood.
Stochastic independence: Two events are called stochastically independent if the probability of their intersection occurring can be decomposed as the product of their individual probabilities.
The above example results in:
The events A and B are therefore not stochastically independent. The team's performance seems to be dependent on the weather.
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