Question

1. given events g and h: p(g) = 0.43; p(h) = 0.26; p(h and g) = 0.14.

(a) make a venn diagram of the situation.
(b) find p(h or g).
(c) find the probability of the complement of event (h and g).
(d) find the probability of the complement of event (h or g).
(e) find the probability of event h, given event g.

1. determine if the following are probability distributions (yes/no). then explain.

(a) x -20 30 40 50
p(x) 0.2 0.3 0.4 0.1
(b) x -5 6 8 10
p(x) 1/7 2/7 2/7 3/7

1. a random variable is a variable that represents a numerical outcome from a probability experiment, you can think of this as numerical data. when we define random variables we let the reader know what the variable represents, including any units of measure. four random variables w, x, y and z are defined below. label each one as either a discrete random variable (d) or a continuous random variable (c).

(a) w = wait time at a drive - thru in minutes
(b) x = number of toppings on a pizza
(c) y = yards for a pass in an nfl football game
(d) z = number of zebras in a zoo

1. a probability distribution table lists all possible values of the random variable along with their associated probabilities. in the table below, x represents the number of students in a 6th grade class in one public school district. if your son will be entering the 6th grade in this district next year, what is the probability that...

x p(x)
32 0.12
33 0.34
34 0.42
35 0.1
36 0.02
a. he will be in a class that has 34 students?
b. he will be in a class that has 37 students?
c. he will be in a class that has fewer than 34 students?
d. he will be in a class that has at least 34 students?

Explanation:

Problem 3(b): Apply OR rule

$P(H \text{ OR } G) = P(G) + P(H) - P(H \text{ AND } G)$
$= 0.43 + 0.26 - 0.14 = 0.55$

Problem 3(c): Complement of intersection

$P(\text{not } (H \text{ AND } G)) = 1 - P(H \text{ AND } G)$
$= 1 - 0.14 = 0.86$

Problem 3(d): Complement of union

$P(\text{not } (H \text{ OR } G)) = 1 - P(H \text{ OR } G)$
$= 1 - 0.55 = 0.45$

Problem 3(e): Conditional probability

$P(H|G) = \frac{P(H \text{ AND } G)}{P(G)}$
$= \frac{0.14}{0.43} \approx 0.326$

Problem 4(a): Check distribution

Sum of P(X): $0.2 + 0.3 + 0.4 + 0.1 = 1$; all P(X) ∈ [0,1]

Problem 4(b): Check distribution

Sum of P(X): $\frac{1}{7} + \frac{2}{7} + \frac{2}{7} + \frac{3}{7} = \frac{8}{7} > 1$

Problem 5(a): Classify W

Wait time is measurable (any value in interval)

Problem 5(b): Classify X

Number of toppings is countable (integer values)

Problem 5(c): Classify Y

Yards for pass is measurable (any value in interval)

Problem 5(d): Classify Z

Number of zebras is countable (integer values)

Problem 6(a): P(X=34)

From table, P(34) = 0.42

Problem 6(b): P(X=37)

37 not in table; P(37) = 0

Problem 6(c): P(X<34)

Sum P(32)+P(33): $0.12 + 0.34 = 0.46$

Problem 6(d): P(X≥34)

Sum P(34)+P(35)+P(36): $0.42 + 0.1 + 0.02 = 0.54$

Answer:

3(b) 0.55
3(c) 0.86
3(d) 0.45
3(e) ≈0.326
4(a) Yes, sum of probabilities is 1 and all are between 0 and 1.
4(b) No, sum of probabilities is greater than 1.
5(a) C
5(b) D
5(c) C
5(d) D
6(a) 0.42
6(b) 0
6(c) 0.46
6(d) 0.54