MATH 211

Final | General
1. An absent-minded nurse is to give Mr. Sean a pill each day. The probability that the nurse forgets to administer the pill is 4/5. If he receives the pill, the probability that Mr. Sean will die is 1/4. If he does not get his pill, the probability that he will die is 1/3.
Mr Sean did not die. What is the probability that the nurse did not forget to give Mr. Sean the pill?

A) $8/205$
B) $28/205$
C) $32/41$
D) $9/41$
E) None given

2. A quiz consists of $10$ multiple choice questions, each with $4$ possible choices. For someone who makes random guesses for all of the questions, find the probability of failing if the minimum passing grade is $70%$.

A) $3.51%$
B) $99.65%$
C) $22.41%$
D) $35.10%$
E) None given

3. Suppose that the random variable is normally distributed with mean value of $10.5$ and standard deviation $1.5$. Find the probability $P(x \le 8.5)$.

A) $91.21%$
B) $40.88%$
C) $9.12%$
D) $4.88%$
E) None given

Free Response

1. Two fair dice are rolled. The sample space consists of the following $36$ outcomes.
S= { (1,1), (1,2), (1,3), (1,4), (1,5), (1,6),
(2,1), (2,2), (2,3), (2,4), (2.5), (2,6),
(3,1), (3,2), (3,3), (3,4), (3,5), (3,6),
(4,1), (4,2), (4,3), (4,4), (4,5), (4,6),
(5,1), (5,2), (5,3), (5,4), (5,5), (5,6),
(6,1), (6,2), (6,3), (6,4), (6,5), (6,6) }
Let $E =$ the event that the sum of the two outcomes is a $6$.
Let $F =$ the event that the larger or the common value of the two outcomes is a $3$.
What is the probability of $E$ given that $F$ has occurred?

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2. A bag full of Halloween candy has $20$ chocolate bars, $4$ licorice candies, and $9$ lollipops, and $10$ pieces of gum totaling in $43$ pieces of candy. Suppose you randomly choose a handful of $5$ pieces of candy.
a) What is the probability that your handful contains exactly $3$ pieces of gum and $2$ chocolate bars?

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b) What is the probability that your handful contains at most $2$ pieces of gum?

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3. Mr Clopu, the Dean of Plopu Community College, drives along a stretch of road with $10$ stoplights along the route. He knows that he has a $0.62$ probability of getting a red light at any one light, and that they are all independent of one another. Let $X =$ the number of red lights he gets. Find the following probabilities.
a. $P(x = 6)$

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b. $P(3 \le x \le 8)$

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c. What is the mean number of green lights he should expect?

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4. A large Zoology class at Plopu Community College had a test whose scores were normally distributed with a mean score of $53$ and a standard deviation of $6$. A test is chosen at random. Find the following probabilities.
a. $P(x \le 70)$

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b. $P(75 \le x \le 85)$

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c. $P(x \ge 90)$

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5. Let $f(x) = 1 - \frac{x}{2}$ for $0 \le x \le 2$ (and assume $f(x) = 0$ for all other $x$).
a. Show that this a probability density function (pdf).

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b. Sketch the pdf, $f(x)$

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c. Find a formula for the cumulative density function (cdf).

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d. Find $P( \frac{1}{2} \le X \le \frac{3}{2} )$.

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