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?

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?

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?

a) What is the probability that your handful contains exactly $3$ pieces of gum and $2$ chocolate bars?

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

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. $P(x = 6)$

a. $P(x = 6)$

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

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. $P(x \le 70)$

a. $P(x \le 70)$

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).

a. Show that this a probability density function (pdf).

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

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

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