Math Problem Statement
Complete the following probability distribution table and then calculate the stated probabilities. Outcome a b c d e Probability 0.2 0.08 0.5 0.02 (a) P({a, c, e}) P({a, c, e}) = (b) P(E ∪ F), where E = {a, c, e} and F = {b, c, e} P(E ∪ F) = (c) P(E ′), where E is as in part (b) P(E ′) = (d) P(E ∩ F ), where E and F are as in part (b) P(E ∩ F) =
Solution
Let's start by completing the probability distribution table.
Step 1: Completing the Probability Distribution Table
The given probabilities are for the outcomes , but we only have probabilities for and . We can find the missing probability for by using the fact that the sum of all probabilities must equal 1.
Given:
- Probability of
- Probability of
- Probability of
- Probability of
The sum of these probabilities is:
Therefore, the probability for must be:
So, the completed probability distribution table is:
Outcome | a | b | c | d | e |
---|---|---|---|---|---|
Probability | 0.2 | 0.08 | 0.5 | 0.02 | 0.2 |
Step 2: Calculating the Stated Probabilities
(a)
This is the probability of the union of the outcomes , , and . The probability of this event is simply the sum of the individual probabilities:
So,
(b) , where and
The union includes all the outcomes in either or , or in both. So: Thus, the probability of is the sum of the probabilities of these outcomes:
So,
(c) , where
The complement consists of the outcomes not in , so: Thus, the probability of is the sum of the probabilities of and :
So,
(d) , where and
The intersection includes the outcomes that are in both and , so: Thus, the probability of is the sum of the probabilities of and :
So,
Summary of Answers:
(a)
(b)
(c)
(d)
Would you like further explanations or details on any of these steps? Here are some related questions for practice:
- What happens if the sum of probabilities exceeds 1 in a probability distribution?
- How do we calculate the probability of the intersection of two disjoint events?
- Can two events have a union that has a probability greater than 1? Why or why not?
- How would the probability distribution change if the probability of outcome was increased to 0.1?
- What does the complement of an event represent in probability theory?
Tip: Always check that the sum of all probabilities in a probability distribution equals 1! This is a fundamental property.
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Math Problem Analysis
Mathematical Concepts
Probability
Union and Intersection of Sets
Complementary Events
Formulas
P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
P(A′) = 1 - P(A)
P(A ∩ B) = sum of probabilities of outcomes common to A and B
Theorems
Addition Rule of Probability
Complement Rule
Intersection of Events
Suitable Grade Level
Grades 7-10
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