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| − | Let <math>A</math> is the event that people like butter <math>\\Rightarrow P(A) = 0.8</math>
| + | <i>A</i>: event that people like butter |
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| | + | <i>B</i>: event that people like jelly |
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| − | '''Probability'''
| + | <math>P(B|A) = \frac{P(A\cap B)}{P(A)} = \frac{0.78}{0.8} = 0.975</math> |
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| − | <br>2-1. | |
| − | Suppose 80% of people like peanut butter, 89% like jelly, and 78% like both. Given that a randomly sampled person likes peanut butter, what is the probability that she also likes jelly?
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| − | [[TDSM 2.1|(Solution 2.1)]]
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| − | <br>2-3.
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| − | Consider a game where your score is the maximum value from two dice. Compute the probability of each event from <math>\{1, \ldots, 6\}</math>
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| − | [[TDSM 2.3|(Solution 2.3)]]
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| − | <br>2-5.
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| − | If two binary random variables <i>X</i> and <i>Y</i> are independent, is <math>\bar{X}</math> (the complement of <i>X</i>) and <i>Y</i> also independent? Give a proof or a counterexample.
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| − | [[TDSM 2.5|(Solution 2.5)]]
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| − | '''Statistics'''
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| − | <br>2-7.
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| − | Construct a probability distribution where none of the mass lies within one <math>\sigma</math> of the mean.
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| − | [[TDSM 2.7|(Solution 2.7)]]
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| − | <br>2-9.
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| − | Show that the arithmetic mean equals the geometric mean when all terms are the same.
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| − | [[TDSM 2.9|(Solution 2.9)]]
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| − | '''Correlation Analysis'''
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| − | <br>2-11.
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| − | What would be the correlation coefficient between the annual salaries of college and high school graduates at a given company, if for each possible job title the college graduates always made:
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| − | <ol type="a">
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| − | <li>5,000 dollars more than high school grads?</li>
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| − | <li>25% more than high school grads?</li>
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| − | <li>15% less than high school grads?</li>
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| − | </ol>
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| − | [[TDSM 2.11|(Solution 2.11)]]
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| − | <br>2-13.
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| − | Use data or literature found in a Google search to estimate/measure the strength of the correlation between:
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| − | <ol type="a">
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| − | <li>Hits and walks scored for hitters in baseball.</li>
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| − | <li>Hits and walks allowed by pitchers in baseball.</li>
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| − | </ol>
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| − | [[TDSM 2.13|(Solution 2.13)]]
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| − | '''Logarithms'''
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| − | <br>2-15.
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| − | Show that the logarithm of any number less than 1 is negative.
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| − | [[TDSM 2.15|(Solution 2.15)]]
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| − | <br>2-17.
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| − | Prove that
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| − | <math>x \cdot y = b^{(\log_b x + \log_b y)}</math>
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