How Do You Calculate the Probability of No Events Occurring?

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In summary: Thus,$$\overline{A \cup B \cup C} = \overline{A} \cap \overline{B} \cap \overline{C},$$so the probability of no events occurring is indeed$$P(\text{none occur}) = P(\overline{A \cup B \cup C}) = P(\overline{A} \cap \overline{B} \cap \overline{C}),$$and you can proceed from there.
  • #1
r0bHadz
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Homework Statement


1.

Suppose that A, B, and C are 3 independent events such that Pr(A)=1/4, Pr(B)=1/3 and Pr(C)=1/2.

a. Determine the probability that none of these events will occur.

Is it just:

(1-P(a))(1-P(b))(1-P(c)) = 3/4 * 2/3 * 1/2 = 1/4

Homework Equations

The Attempt at a Solution


I tried to do 1. another way:

The probability that all theses events will occur: 1/4 * 1/3 * 1/2 = 1/24

1-(1/24) = 23/24

Obviously this is wrong. Is the reason it is wrong, because: the complement of "all of these events will occur" is that "not all of these events will occur," meaning, it is not "none of these events will occur."

None of these events will occur is included in the compliment 1-(1/24), but so is that 1 of the events occur, and that 2 of the events occur, etc.

Am I right in my reasoning?
 
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  • #2
the (1) is correct.
for reference, in general the answer can be found by calculating multinomial distribution.
in (3), the 23/24 probability is sum of "no events", "A only", "B only", "C only", "A&B", "A&C", "B&C".
 
  • #3
r0bHadz said:

Homework Statement


1.

Suppose that A, B, and C are 3 independent events such that Pr(A)=1/4, Pr(B)=1/3 and Pr(C)=1/2.

a. Determine the probability that none of these events will occur.

Is it just:

(1-P(a))(1-P(b))(1-P(c)) = 3/4 * 2/3 * 1/2 = 1/4

Homework Equations

The Attempt at a Solution


I tried to do 1. another way:

The probability that all theses events will occur: 1/4 * 1/3 * 1/2 = 1/24

1-(1/24) = 23/24

Obviously this is wrong. Is the reason it is wrong, because: the complement of "all of these events will occur" is that "not all of these events will occur," meaning, it is not "none of these events will occur."

None of these events will occur is included in the compliment 1-(1/24), but so is that 1 of the events occur, and that 2 of the events occur, etc.

Am I right in my reasoning?

Yes, that's it exactly.
 
  • #4
r0bHadz said:

Homework Statement


1.

Suppose that A, B, and C are 3 independent events such that Pr(A)=1/4, Pr(B)=1/3 and Pr(C)=1/2.

a. Determine the probability that none of these events will occur.

Is it just:

(1-P(a))(1-P(b))(1-P(c)) = 3/4 * 2/3 * 1/2 = 1/4

Homework Equations

The Attempt at a Solution


I tried to do 1. another way:

The probability that all theses events will occur: 1/4 * 1/3 * 1/2 = 1/24

1-(1/24) = 23/24

Obviously this is wrong. Is the reason it is wrong, because: the complement of "all of these events will occur" is that "not all of these events will occur," meaning, it is not "none of these events will occur."

None of these events will occur is included in the compliment 1-(1/24), but so is that 1 of the events occur, and that 2 of the events occur, etc.

Am I right in my reasoning?

You are correct, and there is sound reasoning to justify that fact, as follows. If we denote the complement of any event ##E## as ##\bar{E}##, then
$$
\{ \text{none occur} \} = \overline{A \cup B \cup C},$$
because the event that at least one occurs is ##A \cup B \cup C,## so the complement of that is the event that none occurs.

However, there is a general set-theoretic result:
$$\overline{ \bigcup_{i=1}^n A_i } = \bigcap_{i=1}^n \overline{A_i}$$ That is, the complement of a union is the intersection of the complements.
 

FAQ: How Do You Calculate the Probability of No Events Occurring?

1. What is an event in the context of sample space?

An event is a subset of the sample space, which is the set of all possible outcomes of a random experiment. It represents a specific outcome or a combination of outcomes that we are interested in observing.

2. How do you determine the sample space of an experiment?

The sample space of an experiment can be determined by listing all possible outcomes of the experiment. This can be done by using a tree diagram, a table, or by using the fundamental counting principle.

3. What is the difference between a simple event and a compound event?

A simple event is an event that consists of only one outcome, while a compound event is an event that consists of more than one outcome. Simple events are also known as elementary events.

4. Can the sample space of an experiment change?

Yes, the sample space of an experiment can change if the conditions or parameters of the experiment are altered. For example, if a coin is flipped 10 times, the sample space would consist of 2^10 possible outcomes. But if the coin is flipped only 5 times, the sample space would consist of 2^5 possible outcomes.

5. How is the concept of sample space used in probability?

The sample space is used in probability to determine the likelihood of an event occurring. By calculating the ratio of the number of outcomes in an event to the total number of outcomes in the sample space, we can determine the probability of that event happening.

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