MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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Transcribed Image Text:**Educational Content: Probability Exercise**
**Scenario:**
Your fortune teller’s jar contains 50 cards. Each card is labeled with a predicted theme that will strongly influence how you experience the next year. The number of cards for each of the 4 themes is as follows:
- 10 “lucky”
- 20 “sad”
- 15 “powerful”
- 5 “kind”
**Exercise:**
Determine the probability of drawing the following (leave answers in fractional form):
a. On a single draw, “lucky”.
b. On a single draw, “sad” or “powerful”.
c. Drawing a “sad” and then, after replacing it, a “lucky”.
d. Drawing a sequence of “lucky”, “kind”, and “kind” (without replacement).
**Explanation:**
- **Probability of a Single Event:**
The probability of drawing a card labeled “lucky” on a single draw is calculated by dividing the number of “lucky” cards by the total number of cards.
- **Probability of Combined Events:**
For drawing “sad” or “powerful”, add the probabilities of each. This involves combining the counts of “sad” and “powerful” cards over the total card count.
- **Probability with Replacement:**
When drawing a “sad” card, replacing it, and then drawing a “lucky”, consider the probability of each event separately and then multiply them. The total card number remains constant due to replacement.
- **Probability without Replacement:**
When drawing a specific sequence like “lucky”, “kind”, and “kind” without replacing the cards, adjust the total card count after each draw and multiply the probabilities of each step. This takes into account the changing deck size.
![**Sample Mean:**
\[
\bar{X} = \frac{\Sigma X}{n}
\]
Where:
- \(\Sigma\) = sum (or add up)
- \(\bar{X}\) = sample mean
- \(n\) = total number of scores, or the sample size
**For a population**, mean is represented by the Greek letter mu (\(\mu\)).
---
**Population Variance:**
\[
\sigma^2 = \frac{\Sigma (X - \mu)^2}{N}
\]
**Sample Variance:**
\[
s^2 = \frac{\Sigma (X - \bar{X})^2}{n - 1}
\]
Remember that the **standard deviation** (\(\sigma\) or \(s\)) is the square root (\(\sqrt{\ }\)) of the variance.
---
**Z Score (for one score/individual):**
\[
Z = \frac{X - \mu}{\sigma}
\]
**Z Score: One Sample Z test:**
\[
Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}
\]
\(\bar{X}\) = mean of sample
\(\mu\) = mean of population
\(\sigma\) = standard deviation of population
\(n\) = sample size
\(\sqrt{n}\) = square root of sample size](https://content.bartleby.com/qna-images/question/7dee35c8-07bc-4f6f-8047-1c0be93c606e/af285d6d-b045-4c88-928f-3bad823923bc/l7e3wcl_processed.jpeg)
Transcribed Image Text:**Sample Mean:**
\[
\bar{X} = \frac{\Sigma X}{n}
\]
Where:
- \(\Sigma\) = sum (or add up)
- \(\bar{X}\) = sample mean
- \(n\) = total number of scores, or the sample size
**For a population**, mean is represented by the Greek letter mu (\(\mu\)).
---
**Population Variance:**
\[
\sigma^2 = \frac{\Sigma (X - \mu)^2}{N}
\]
**Sample Variance:**
\[
s^2 = \frac{\Sigma (X - \bar{X})^2}{n - 1}
\]
Remember that the **standard deviation** (\(\sigma\) or \(s\)) is the square root (\(\sqrt{\ }\)) of the variance.
---
**Z Score (for one score/individual):**
\[
Z = \frac{X - \mu}{\sigma}
\]
**Z Score: One Sample Z test:**
\[
Z = \frac{\bar{X} - \mu}{\sigma / \sqrt{n}}
\]
\(\bar{X}\) = mean of sample
\(\mu\) = mean of population
\(\sigma\) = standard deviation of population
\(n\) = sample size
\(\sqrt{n}\) = square root of sample size
Expert Solution
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Step 1
Since you have posted multiple sub-parts, we will solve the first three sub-parts for you. To get the remaining sub-parts kindly repost the complete question and mention the subparts that need to be solved.
Given,
Lucky cards = 10
Sad cards = 20
Powerful cards = 15
Kind cards = 5
Total number of cards = 50
Step by stepSolved in 3 steps
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