Addition Rule of Probability: Example Problems

You spin a spinner that has 8 equal-sized sections numbered 1 to 8. Find the theoretical probability of landing on the given section(s) of the spinner. (i) section 1 (ii) odd-numbered section (iii) a section whose number is a power of 2. [4 MARKS]

Write additive inverse and multiplicative inverse of $\frac{1}{5}$.

Use the special addition rule to determine the probability of drawing either a spade OR a heart from a standard deck of cards, on one draw from the deck.

There is 16 white, 9 black and 7 yellow tulips. What's the probability of randomly picking 9 flowers where 2 are white, 3 are black, and 4 are yellow?

What is the key word that indictes that the Addition Rule for Probabilities will be used?

What is another name for mutually exclusive events?

How do you add probabilities?

The probability that John falls off a ladder and breaks his arm is 0.2, and the probability that John falls off a ladder and breaks his leg is .08. Are these events mutually exclusive (disjoint)? Why or why not?

Could we use the special addition rule for determining the probability that for one draw from a deck of cards, that the card is either a Queen or a Heart? Why or why not?

In a jar are 10 marbles: 4 black, 5 blue, and 1 yellow. Why can we use the special addition rule to calculate the probability that a single marble drawn from the jar will be either black or yellow? What is the probability?

What is the probability that, in a single draw from a standard deck of cards, we will get either a Jack or a Diamond?

You roll two balanced dice one time. What is the probability that you obtain either a sum of 8 or the same number on each of the two dice?

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?

What is the addition rule for mutually exclusive events?

What is the difference between events that are mutually exclusive and those that are not mutually exclusive?

A game is played by rolling a six sided die which has four red faces and two blue faces. One turn consists of throwing the die repeatedly until a blue face is on top or the die has been thrown 4 times
Adnan and Beryl each have one turn. Find the probability that Adnan throws the die more turns than Beryl
I tried : Adnan throws two times and Beryl throws once = $\frac{2}{3}$ x $\frac{1}{3}$
Adnan throws three times and Beryl throws once = $\frac{4}{9}$ x $\frac{1}{2}$
Adnan throws three times and Beryl throws twice = $\frac{4}{9}$ x $\frac{2}{3}$
Adnan throws four times and Beryl throws once = $\frac{8}{27}$ x $\frac{1}{2}$
Adnan throws four times and Beryl throws twice = $\frac{8}{27}$ x $\frac{2}{3}$
Adnan throws four times and Beryl throws three times = $\frac{8}{27}$ x $\frac{4}{9}$
The answer says 0.365

If the probability that Joe will buy a pizza is 0.5 and the probability that Elaine will buy a pizza is 0.35, then what is the probability that at least one of the two will buy a pizza on their next visit to the pizza place?
I know that this is a very basic problem. However, I am a little confused about the "at least" statement in the question. I would really like to understand why the correct answer is correct, so please go in depth for how you got your answer. I am thinking that you might use the addition rule for probability, but again, the "at least" confuses me.

I was thinking today that if something with a probability of occurring of 1% happened 100 times, then the probability of that something happening is 100%, I believe that according to the addition rule for probabilities the probabilities for each event should be added up to get the total probability thus 1/100 + 1/100 + 1/100 ... up to 100 = 100/100 = 1 = 100%.

Now, there's still the possibility that the event didn't occur any one of those 100 times when it could have, because each time is independent. If such is the case, then obviously the probability is not 100%.

I believe I'm wrong and that I'm doing something wrong. So I would very much appreciate any guidance as to how to go about calculating the probability of something that happens 100 times that has a chance of occurring of 1% every time. For example, let's say there's a probability of 1% of dying from eating too much Cap'n Crunch, if I ate too much Cap'n Crunch 100 times, what is the probability that I will die?

A coin is tossed three times: The probability of zero heads is 1/8 and the probability of zero tails is 1/8.

And my question is: What is the probability that all three tosses result in the same outcome?

So, if P(zero heads)= 1/8 , then that should be the same of p(all tails)?

If so, we could use the Addition Rule which is $P(A\cup <!--\; \cup \; -->B)=P(A)+P(B)$

where A and B are disjoint events, i.e. A$A\cap <!--\; \cap \; -->B=\mathrm{\varnothing <!--\; \varnothing \; -->}$, A is the event of tossing all heads and B is the event of tossing all tails.

I'm not sure how to continue after that... would the complement be used?