# Manufacturing plants probability problem. Please for help!

#### fuUuUzZzZy

##### New member
Hey guys. I'm new here, and I need some help if there is possible.

The problem is:

Tree manufacturing plants produce similar products. The fist one produce 15%, the second one 55% and third the rest of the whole production. Plan 1 produces 3% defective products, plan 2 produces 4% and plant 3 produces 5% defective products. A product is selected at random.

a) What is the probability that it will not be defective? What is the probability it came from the second plant?

b) Find the probability of the event that the selected product is defective.

Thank You,
Vlatko

#### soroban

##### Elite Member
Hello, fuUuUzZzZy!

Tree manufacturing plants produce similar products.
The first one produce 15%, the second one 55% and third the rest of the whole production.
Plant A produces 3% defective products, plant B produces 4% and plant C produces 5% defective products.

$$\displaystyle \begin{array}{cccc} & \text{Prod'n} & P(\text{def}) & P(\text{good}) \\ \hline \text{plant A} & 15\% & 3\% & 97\% \\ \text{plant B} & 55\% & 4\% & 96\% \\ \text{plant C} & 30\% & 5\% & 95\% \\ \hline\end{array}$$

A product is selected at random.
(a) What is the probability that it will not be defective?

$$\displaystyle \begin{array}{ccccc}P(A\wedge\text{ good}) &=& (0.15)(0.97) &=& 0.1455 \\ P(B\wedge \text{good}) &=& (0.55)(0.96) &=& 0.5280 \\ P(C\wedge\text{good}) &=& (0.30)(0.95) &=& 0.2850 \\ \hline && \text{Total} && 0.9585 \end{array}$$

Therefore: .$$\displaystyle P(\text{good}) \:=\:0.9585$$

(b) What is the probability it came from plant B?

$$\displaystyle P(\text{plant B}\,|\,\text{good}) \:=\:\dfrac{P(\text{plant B}\,\wedge\,\text{good})}{P(\text{good})} \:=\: \dfrac{0.528}{0.9585} \;\approx\;0.55$$

(c) Find the probability of the event that the selected product is defective.

$$\displaystyle P(\text{def}) \;=\;1 - P(\text{good}) \;=\;1 - 0.9585 \;=\;0.0415$$

#### fuUuUzZzZy

##### New member
WOW... That was fast! Thanks a LOT!

I have one problem left.. I solved everything else, except this easy one.

A Lottery consists of 300 lottery tickets. There are 20 tickets with some prize and only two of them with grand prize. Alex bought 12 tickets. Find the probability that he won at least one grand prize if he won 5 prizes.

I think that whole problem can be solved only with one formula:

 Number of ways it can happen Total number of outcomes

But, I'm missing something. Maybe because I'm really tired with all of the exams..

Best Regards,
fuUuUzZzZy

#### soroban

##### Elite Member
Hello, fuUuUzZzZy!

The formula you gave applies to ALL probability problems.
This problem involves Conditional Probability.
If you're not familiar with it, you should not be assigned this problrem.

A lottery consists of 300 lottery tickets.
There are 20 tickets with regular prizes and only two of them with a Grand Prize.
Alex bought 12 tickets.
Find the probability that he won at least one Grand prize, given that he won 5 prizes.

#### mmm4444bot

##### Super Moderator
Staff member
WOW... That was fast! Thanks a LOT!
Did you learn anything from Soroban's camera-ready copy?

#### fuUuUzZzZy

##### New member
Did you learn anything from Soroban's camera-ready copy?
Honestly, I'm very[FONT=arial, sans-serif] [/FONT]grateful for this help. The problem is that I don't have any free time (College, work, family), and the deadline is Thursday. I don't have to go to exam, If I solve this problems like a project. Trust me if I had a one hour free time, I would learn this the best that I can.

I already solved 13 other problems, and those 2 are my last (right now is only one).

Because of that, I'm searching for answer on this forum (because this forum is the best for math help ), I just want to pass this exam (A really don't like math a lot , and my studying field is different then math ).

So guys, If you want to help me (and if the writing answer is not hard for you), please. I will be eternally grateful whole life

Regards,
fuUuUzZzZy

#### mmm4444bot

##### Super Moderator
Staff member
I don't have to go to exam, If I solve this problems like a project.
I'm thinking that your instructor has a different project in mind.

I wish you good fortune in your future career which uses neither math nor logic.

#### Subhotosh Khan

##### Super Moderator
Staff member
And we wonder why all the jobs are going overseas .....!!!!

#### fuUuUzZzZy

##### New member
Ok guys.. don't yell at me

I do what you said before. I tried to solve, and this is what I'v got:

[SUB]280[/SUB]C[SUB]7[/SUB] * [SUB][SUP]18[/SUP][/SUB]C[SUB]4[/SUB] * [SUB]2[/SUB]C[SUB]1[/SUB] + [SUB]280[/SUB]C[SUB]7[/SUB] * [SUB]18[/SUB]C[SUB]3[/SUB] * [SUB]1[/SUB]c[SUB]2[/SUB]
_____________________________________________________
[SUB]300[/SUB]C[SUB]12

At least, can you tell me if it is okay?

Tnx.[/SUB]