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Chapter 4: Problem 3

A sample space contains 10 simple events: \(E_{1}\), \(E_{2}, \ldots, E_{10} .\) If \(P\left(E_{1}\right)=3 P\left(E_{2}\right)=.45\) and the remaining simple events are equiprobable, find the probabilities of these remaining simple events.

Short Answer

Expert verified
Answer: The probability of the remaining simple events (E_3 to E_10) is 0.05.

Step by step solution

01

Write down the given information

We know the following facts: 1. \(P(E_1) = 3P(E_2) = 0.45\) 2. \(P(E_3) = P(E_4) = \ldots = P(E_{10})\) 3. \(P(E_1) + P(E_2) + P(E_3) + \ldots + P(E_{10}) = 1\)
02

Find the probability of \(E_2\)

We know that \(P(E_1) = 3P(E_2) = 0.45\). Divide both sides by 3 to find \(P(E_2)\): $$P(E_2) = \frac{0.45}{3}$$ $$P(E_2) = 0.15$$
03

Write the equation for the sum of all probabilities

The sum of all probabilities is equal to 1, so we can write $$P(E_1) + P(E_2) + P(E_3) + \ldots + P(E_{10}) = 1$$ We know \(P(E_1)\) and \(P(E_2)\), and since \(P(E_3) = P(E_4) = \ldots = P(E_{10})\), we can write this as: $$0.45 + 0.15 + 8P(E_3) = 1$$
04

Solve for \(P(E_3)\)

Now, we'll solve for \(P(E_3)\): $$8P(E_3) = 1 - 0.45 - 0.15$$ $$8P(E_3) = 0.4$$ $$P(E_3) = \frac{0.4}{8}$$ $$P(E_3) = 0.05$$
05

State the probabilities of the remaining simple events

We found that \(P(E_3) = 0.05\), and since all remaining events are equiprobable, we conclude that: $$P(E_4) = P(E_5) = \ldots = P(E_{10}) = 0.05$$

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Most popular questions from this chapter

Gregor Mendel was a monk who suggested in 1865 a theory of inheritance based on the science of genetics. He identified heterozygous individuals for flower color that had two alleles (one \(\mathrm{r}=\) recessive white color allele and one \(\mathrm{R}=\) dominant red color allele). When these individuals were mated, \(3 / 4\) of the offspring were observed to have red flowers and \(1 / 4\) had white flowers. The table summarizes this mating; each parent gives one of its alleles to form the gene of the offspring. We assume that each parent is equally likely to give either of the two alleles and that, if either one or two of the alleles in a pair is dominant (R), the offspring will have red flowers. a. What is the probability that an offspring in this mating has at least one dominant allele? b. What is the probability that an offspring has at least one recessive allele? c. What is the probability that an offspring has one recessive allele, given that the offspring has red flowers?

An experiment can result in one of five equally likely simple events, \(E_{1}, E_{2}, \ldots, E_{5} .\) Events \(A, B,\) and \(C\) are defined as follows: $$A: E_{1}, E_{3} \quad P(A)=.4$$ $$B: E_{1}, E_{2}, E_{4}, E_{5} \quad P(B)=.8$$ $$C: E_{3}, E_{4} \quad P(C)=.4$$. Find the probabilities associated with these compound events by listing the simple events in each. a. \(A^{c}\) b. \(A \cap B\) c. \(B \cap C\) d. \(A \cup B\) e. \(B \mid C\) f. \(A \mid B\) g. \(A \cup B \cup C\) h. \((A \cap B)^{c}\)

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Only \(40 \%\) of all people in a community favor the development of a mass transit system. If four citizens are selected at random from the community, what is the probability that all four favor the mass transit system? That none favors the mass transit system?

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