Probability Theory Example Problem

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Questions and Answers

What is the formula for calculating the probability of the difference between two events A and B, P(B - (A ∩ B))?

P(A ∪ B) - P(A ∩ B)

P(B) - P(A)

P(B) - P(A ∩ B) (correct)

P(A) - P(B ∩ A)

In the given example, if 70% of students are members of a student association (A), 40% are members of a sports club (B), and 20% are members of both, what is P(A ∪ B)?

0.7

0.9 (correct)

0.4

0.5

What is the general formula for calculating the probability of the union of n events A1, A2, ..., An using the inclusion-exclusion principle?

$\prod_{i=1}^{n} P(A_i) - \sum_{1 \leq i < j \leq n} P(A_i \cup A_j) + \sum_{1 \leq i < j < k \leq n} P(A_i \cup A_j \cup A_k) - \cdots$

$\sum_{i=1}^{n} P(A_i) - \sum_{1 \leq i < j \leq n} P(A_i \cap A_j) + \sum_{1 \leq i < j < k \leq n} P(A_i \cap A_j \cap A_k) - \cdots$ (correct)

$\prod_{i=1}^{n} P(A_i) - \sum_{1 \leq i < j \leq n} P(A_i \cap A_j) + \sum_{1 \leq i < j < k \leq n} P(A_i \cap A_j \cap A_k) - \cdots$

$\sum_{i=1}^{n} P(A_i) - \sum_{1 \leq i < j \leq n} P(A_i \cup A_j) + \sum_{1 \leq i < j < k \leq n} P(A_i \cup A_j \cup A_k) - \cdots$

What is the relationship between the events A and B in the formula P(A ∪ B) = P(A) + P(B) - P(A ∩ B)?

No specific relationship is required Signup and view all the answers

If P(A) = 0.4, P(B) = 0.6, and P(A ∩ B) = 0.2, what is P(A ∪ B)?

0.8 Signup and view all the answers

What is the probability of the difference between two events A and B, P(A - B), in terms of the given probabilities P(A), P(B), and P(A ∩ B)?

P(A) - P(A ∩ B) Signup and view all the answers

Given two events A and B, what is the probability of the event A - B (the difference between A and B)?

P(A) - P(B) Signup and view all the answers

If A is a subset of B, which of the following statements is true?

P(A) < P(B) Signup and view all the answers

What is the Inclusion-Exclusion Principle used for?

Finding the probability of the union of more than two events Signup and view all the answers

In the example given, if 70% of a population practices some sport and 10% plays tennis, what is the probability that a random person practices a sport but does not play tennis?

0.6 Signup and view all the answers

What is the formula for calculating the probability of the union of two events A and B?

P(A ∪ B) = P(A) + P(B) - P(A ∩ B) Signup and view all the answers

If A and B are two mutually exclusive events, what is the value of P(A ∩ B)?

0 Signup and view all the answers

What is the probability of the difference between events A and B if A is a subset of B and P(B) = 0.7, P(A) = 0.3?

0.4 Signup and view all the answers

If the probability of events A, B, and C are 0.4, 0.3, and 0.2 respectively, what is the probability of the union of A and B but not C?

0.4 Signup and view all the answers

In the given example, if the probability of intersection between events A and B is 0.2 and P(B) = 0.6 while P(A) = 0.4, what is the probability of the union of A and B?

0.8 Signup and view all the answers

If P(A) = 0.5, P(B) = 0.4, and P(A ∩ B) = 0.1, what is the probability of the difference between events A and B?

0.3 Signup and view all the answers

If a fair dice is rolled four times, what is the probability that at least one roll does not result in a six?

$rac{1295}{1296}$ Signup and view all the answers

If P(A) = 0.3, P(B) = 0.4, and P(A ∩ B) = 0.15, what is the probability of the union of A and B?

$P(A) + P(B) - P(A ∩ B)$ Signup and view all the answers

Study Notes

The formula for calculating the probability of the union of two events is given by Theorem 1.6: P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
In the example provided with a group of students, where 70% are in a student association, 40% in a sports club, and 20% in both, the probability that a random student is in a student association or a sports club is 0.9.
The Inclusion-Exclusion Principle is used to calculate the probability of the union of n events in a common sample space, following the formula: P(A1 ∪ A2 ∪ ... ∪ An) = Σ P(Ai) - Σ P(Ai ∩ Aj) + ... + (-1)^(n+1) P(A1 ∩ A2 ∩ ... ∩ An).
The concept of the complement of an event is explored, where for any event A, P(A) = 1 - P(Ac), using Axiom 3 and Axiom 2 (P(S) = 1).
The relationship between two sets A and B in terms of probability is established by Theorem 1.5: If A is a subset of B, then P(B - A) = P(B) - P(A).

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Description

Solve a probability theory example problem involving the union of two events and the set theory formula. Calculate the probability of a random student being a member of different associations based on given information.