A First Course in Probability (10th Edition)
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ISBN: 9780134753119
Author: Sheldon Ross
Publisher: PEARSON
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![### Problem Statement:
#### Astrobiology and Probability
---
Suppose that there is a 0.9% chance a planet will develop life, a 2.3% chance that the life will evolve into complex organisms, a 7% chance that complex organisms develop into advanced civilizations, and an X% chance that the advanced civilizations become intergalactic.
If there are 2,131,984 planets, and we observe only 1 intergalactic civilization, what is the expected value of X?
---
**Note:**
- X is in percentage form (e.g., 30.51 not 0.3051).
- You can round to two decimal places, but do not round until your final answer!
---
### Explanation:
Consider each step as a sequential probability process:
1. **Probability that a planet develops life:**
\( P(\text{Life}) = 0.009 \)
2. **Probability that life evolves into complex organisms, given that life has already developed:**
\( P(\text{Complex Organisms}|\text{Life}) = 0.023 \)
3. **Probability that complex organisms develop into advanced civilizations, given that complex organisms have developed:**
\( P(\text{Advanced Civilizations}|\text{Complex Organisms}) = 0.07 \)
4. **Probability that advanced civilizations become intergalactic, given that advanced civilizations exist (this is X, which we need to find):**
\( P(\text{Intergalactic}|\text{Advanced Civilizations}) = \frac{X}{100} \)
### Formulating the Combined Probability:
The combined probability for one planet to reach the intergalactic civilization stage is:
\[
P(\text{Intergalactic}) = P(\text{Life}) \times P(\text{Complex Organisms}|\text{Life}) \times P(\text{Advanced Civilizations}|\text{Complex Organisms}) \times P(\text{Intergalactic}|\text{Advanced Civilizations})
\]
### Setting Up the Equation:
With 2,131,984 planets and 1 observed intergalactic civilization:
\[
1 = 2,131,984 \times 0.009 \times 0.023 \times 0.07 \times \frac{X}{100}
\]
\[
1 =](https://content.bartleby.com/qna-images/question/6387f1fe-f7de-460c-a439-51c8967659dd/9057c5ef-6a74-4d28-b7d6-654b423cc84d/31lfyoc_processed.jpeg)
Transcribed Image Text:### Problem Statement:
#### Astrobiology and Probability
---
Suppose that there is a 0.9% chance a planet will develop life, a 2.3% chance that the life will evolve into complex organisms, a 7% chance that complex organisms develop into advanced civilizations, and an X% chance that the advanced civilizations become intergalactic.
If there are 2,131,984 planets, and we observe only 1 intergalactic civilization, what is the expected value of X?
---
**Note:**
- X is in percentage form (e.g., 30.51 not 0.3051).
- You can round to two decimal places, but do not round until your final answer!
---
### Explanation:
Consider each step as a sequential probability process:
1. **Probability that a planet develops life:**
\( P(\text{Life}) = 0.009 \)
2. **Probability that life evolves into complex organisms, given that life has already developed:**
\( P(\text{Complex Organisms}|\text{Life}) = 0.023 \)
3. **Probability that complex organisms develop into advanced civilizations, given that complex organisms have developed:**
\( P(\text{Advanced Civilizations}|\text{Complex Organisms}) = 0.07 \)
4. **Probability that advanced civilizations become intergalactic, given that advanced civilizations exist (this is X, which we need to find):**
\( P(\text{Intergalactic}|\text{Advanced Civilizations}) = \frac{X}{100} \)
### Formulating the Combined Probability:
The combined probability for one planet to reach the intergalactic civilization stage is:
\[
P(\text{Intergalactic}) = P(\text{Life}) \times P(\text{Complex Organisms}|\text{Life}) \times P(\text{Advanced Civilizations}|\text{Complex Organisms}) \times P(\text{Intergalactic}|\text{Advanced Civilizations})
\]
### Setting Up the Equation:
With 2,131,984 planets and 1 observed intergalactic civilization:
\[
1 = 2,131,984 \times 0.009 \times 0.023 \times 0.07 \times \frac{X}{100}
\]
\[
1 =
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