High school probability Uncovered: Discover the Secrets with Plainmath's Expert Help

In a computer game, when choosing a character there are 5 choices for hair style, 3 choices for eye color, and 7 choices for outfits. How many different characters are possible if you were to choose 1 hair style, 1 eye color, and 1 outfit?

Events $A,B,C$ are such that $B$ and $C$ are mutually exclusive and $P(A)=2/3,P(A\cup <!--\; \cup \; -->B)=5/6$ and $P(B\cup <!--\; \cup \; -->C)=4/5$. If $P(B|A)=1/2$ and $P(C|A)=3/10$, calculate $P(C)$.

In a​ state's Pick 3 lottery​ game, you pay ​$1.27 to select a sequence of three digits​ (from 0 to​ 9), such as 244. If you select the same sequence of three digits that are​ drawn, you win and collect ​$468.14.
a. How many different selections are​ possible?
b. What is the probability of​ winning?
c. If you​ win, what is your net​ profit? ​
d. Find the expected value.
e. If you bet $1.27 in a certain​ state's Pick 4​ game, the expected value is negative −$0.80. Which bet is​ better, a ​$1.27 bet in the Pick 3 game or a $ 1.27 bet in the Pick 4​ game? Explain.
A. The Pick 4 game is a better bet because it has a larger expected value.
B. The Pick 3 game is a better bet because it has a larger expected value.
C. Neither bet is better because both games have the same expected value.

If $P(A)=1/3$ and $P(B^{\mathrm{\complement <!--\; \complement \; -->}})=1/4$, then, can $A$ and $B$ be mutually exclusive?

The Porsche Club of America sponsors driver education events that provide high-performance driving instruction on actual racetracks. Because safety is a primary consideration at such events, many owners elect to install roll bars in their cars. Deegan Industries manufactures two types of roll bars for Porsches. Model DRB is bolted to the car using existing holes in the car's frame. Model DRW is a heavier roll bar that must be welded to the car's frame. Model DRB requires 20 pounds of a special high alloy steel, 40 minutes of manufacturing time, and 60 minutes of assembly time. Model DRW requires 25 pounds of the special high alloy steel, 100 minutes of manufacturing time, and 40 minutes of assembly time. Deegan's steel supplier indicated that at most 40,000 pounds of the high-alloy steel will be available next quarter. In addition, Deegan estimates that 2000 hours of manufacturing time and 1600 hours of assembly time will be available next quarter. The pro?t contributions are $200 per unit for model DRB and $280 per unit for model DRW. The linear programming model for this problem is as follows:
Max 200DRB + 280DRW
s.t.
20DRB + 25DRW 40,000 Steel Available
40DRB + 100DRW ? 120,000 Manufacturing minutes
60DRB + 40DRW ? 96,000 Assembly minutes
DRB, DRW ? 0
Optimal Objective Value = 424000.00000
Variable Value blackuced Cost
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 1000.00000 0.00000
DRW 800.00000 0.00000
Constraint Slack/ Surplus Dual Value
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 0.00000 8.80000
2 0.00000 0.60000
3 4000.00000 0.00000
Objective Allowable Allowable
Variable Coef?cient Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
DRB 200.00000 24.00000 88.00000
DRW 280.00000 220.00000 30.00000
RHS Allowable Allowable
Constraint Value Increase Decrease
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
1 40000.00000 909.09091 10000.00000
2 120000.00000 40000.00000 5714.28571
3 96000.00000 Infnite 4000.00000
a. What are the optimal solution and the total profit contribution?
b. Another supplier offeblack to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.
c. Deegan is considering using overtime to increase the available assembly time. What would you advise Deegan to do regarding this option? Explain.
d. Because of increased competition, Deegan is considering blackucing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain.
e. If the available manufacturing time is increased by 500 hours, will the dual value for the manufacturing time constraint change? Explain.

When do you add probabilities?

Let X be the set of 10-digit numbers that do not contain all digits 0 through 9 in their decimal representation and do not begin with 0. How many elements does the set X have?
I think this should be done by counting principle. So I have 10 spots, and 10 options for each spot. However, the first number can't be 0, so the first spot has 9 options.
9 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 9
I think the 10th spot only have 9 options since I can't repeat one number (the representation doesn't have all the digits).

The average height of U.S. men is 69 inches (5 feet 9 inches) with a standard deviation of 4 inches. What is the z-value that corresponds to the shortest 33% of men?
Select one:
a. -0.95
b. -0.13
c. -0.33
d. -0.44
e. 0.95
f. 0.44
g. 0.33
h. 0.13

Assume that​ women's heights have a mean of 63 inches ​(5 feet 3​inches) and a standard deviation of 3 inches. a. What​ women's height corresponds to a​ z-score of 1.20​? b. What​ women's height corresponds to a​ z-score of negative −1.00​?

Let $\mathcal{A}$=event space. Assume that $\{A_n\}$ is a monotone nondecreasing sequence of events in A$\mathcal{A}$. Let $\{A_n\}$ be a sequence of sets defined as follows:
$B_1=A_1;B_i=A_i-<!--\; -\; -->A_{i-<!--\; -\; -->1}\text{for}i\in <!--\; \in \; -->\{2,3,4,\dots <!--\; \dots \; -->\}$
Prove that $\{B_n\}$ is a sequence of mutually exclusive events in $\mathcal{A}$. Justify each line of proof.
I know that what I need to show here is $B_i\cap <!--\; \cap \; -->B_j=\mathrm{\varnothing <!--\; \varnothing \; -->}$ for any $i<j$. I've divided it to be proof by cases where case $1$ is $i=1\text{and}j\in <!--\; \in \; -->\{2,3,4,\dots <!--\; \dots \; -->\}$ and case $2$ is $i\in <!--\; \in \; -->\{2,3,4,\dots <!--\; \dots \; -->\}$ and $j\in <!--\; \in \; -->\{2,3,4,\dots <!--\; \dots \; -->\}$ where $i<j$ so that $B_i=A_i-<!--\; -\; -->A_{i-<!--\; -\; -->1}\text{and}{B}_j=A_j-<!--\; -\; -->A_{j-<!--\; -\; -->1}$ and $B_i=A_i-<!--\; -\; -->A_{i-<!--\; -\; -->1}\text{and}{B}_j=A_j-<!--\; -\; -->A_{j-<!--\; -\; -->1}$ where $i\in <!--\; \in \; -->\{2,3,4\dots <!--\; \dots \; -->\}$ and $j\in <!--\; \in \; -->\{i+1,i+2,\dots <!--\; \dots \; -->\}$.

The masses of oil palm sold in container have a normal distribution with average of 1,000kg and standard deviation of 35kg. Find the probability that product will weight
a) Less than 978kg
b) Less than 1,027kg
c) Between 978kg and 1,027kg

$A\cup <!--\; \cup \; -->B$ can be expressed as the union of two mutual exclusive events in the following way:
$A\cup <!--\; \cup \; -->B=A\cup <!--\; \cup \; -->(B-<!--\; -\; -->AB)$
Express $A\cup <!--\; \cup \; -->B\cup <!--\; \cup \; -->C$ in a similar way.

How do you use the Counting Principle to find the probability of rolling a 4 on each of 4 number cubes?

Find an $x$ such that mutually exclusive events are to disjoint sets what independent events are to $x$

Events $A$ and $B$ are independent, events $A$ and $C$ are mutually exclusive, and events $B$ and $C$ are independent.
If $P(A)=12;P(B)=14;P(C)=18$; what is $P(A\cup B\cup C)$?

PNC Bank agency receives in average 20 customers per hour. Due to the current capacity, this agency has an average service time of 30 minutes every hour. The manager would like you to perform the following analysis.
What is the probability of finding a busy system?
Expected number of customers in the system
Expected number of customers waiting in the system

What is the addition rule for mutually exclusive events?