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**Probabilty**

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**caramac****Guest**

Hi everyone im new to this forum my name is cam. My dad needs help on one of his assignment questions would be grateful for some answers.

cheers cam.

This is the question

Your experience has shown that the ready-mixed concrete supplier Everconc delivers its order on time with probability one half, one day late with probability one quarter, and two days late also with probability one quarter.

A second supplier, quite unrelated to the concrete supplier, called Steelfix, delivers its order on time with probability two thirds, one date late with probability one sixth, and two days later with probability one sixth.

To complete your project according to your deadlines, you need both suppliers to deliver on time. A delay by either of them causes you to miss your deadline by a corresponding amount. Thus, for example, if the Everconc supplier is on time but Steelfix is late by one day, you are still late by one day.

Barring any other sources of possible delay, what is the probability that you will complete your project:

1. on time?

2. one day late?

3. two days late?

A third independent company can supply both concrete and re-bar steel such that they arrive together and are equally likely to be on time, one day late or two days late. From a probabilistic viewpoint, assuming that the combined costs of materials to your company are the same, do you prefer to keep Everconc and Steelfix or switch to the third company that supplies both?

Explain your answer and mention any advantages or disadvantages of having a sole supplier.

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 20,754

Supply by the first two suppliers:-

Required probability

On time = 1/3

One day late = 1/4+1/6=5/12

Two days late = 1/4

Since the costs are the same, the supply by two suppliers has a higher probability of being in time or one day late than supply by a single supplier. Hence, I would prefer Everconc and Steelfix to the third company. Thats what I think

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**mathsyperson****Moderator**- Registered: 2005-06-22
- Posts: 4,900

I got probabilities of:

On time: 1/3

1 day late: 7/24

2 days late: 9/24

And then by using the same reasoning as ganesh, but the other way around, it would be better to switch to the 3rd company.

Why did the vector cross the road?

It wanted to be normal.

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**yonski****Member**- Registered: 2005-12-14
- Posts: 67

Yeah, i agree with Mathsyperson. Think about it this way:

**On time:**

The only way for the work to be completed on time is for both supliers to be on time (0 days late). This gives the one possibility

0 and 0 = 1/2 × 2/3 = 1/3.

**One day late:**

There are three possible ways for the work to be completed one day late. Either both suppliers are a day late, or the first is on time and the second is a day late, or the first is a day late and the second is on time. This gives:

1 and 1 = 1/4 × 1/6 = 1/24

0 and 1 = 1/2 × 1/6 = 2/24

1 and 0 = 1/4 × 2/3 = 4/24

Therefore the overall probability is 1/24 + 2/24 + 4/24 = 7/24.

**Two days late:**

Using similar reasoning as before we have five possible ways in which the work can be completed two days late. These are:

2 and 2 = 1/4 × 1/6 = 1/24

1 and 2 = 1/4 × 1/6 = 1/24

2 and 1 = 1/4 × 1/6 = 1/24

0 and 2 = 1/2 × 1/6 = 2/24

2 and 0 = 1/4 × 2/3 = 4/24

This gives an overall probability of 1/24 + 1/24 + 1/24 + 2/24 + 4/24 = 9/24.

Student: "What's a corollary?"

Lecturer: "What's a corollary? It's like when a theorem has a child. And names it corollary."

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 20,754

I agree. mathsyperson is right! I thought about it after posting; the probailities for one day and two days late given by mathsy are correct!

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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