Probability Question I have got wrong? "5% of the vases, the colour is uneven...."

mcarthyryan

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Probability Question I have got wrong? "5% of the vases, the colour is uneven...."

The question:

A company producing glass vases is experiencing quality issues and has identified the following independent problems.

5% of the vases the colour is uneven.
6% of the vases the glaze is damaged.
4% of the vases have a distorted shape.

What is the overall probability of a vase having one or more of these defects?

Give your answer as a percentage to 1 decimal place.

My answer:

Probability of a vase of having uneven colour, P(C) = 5% = 1/20
Probability of a vase of having a damaged glaze, P(G) = 6% = 3/50
Probability of a vase of having a distorted shape, P(S) = 4% = 1/25

P (≥1 defects)
Probability of having 1 defect: P(1) = P (uneven colour) or P (damaged glaze) or P (distorted shape)
P(1) = 1/20 + 3/50 + 1/25 = 3/20

Probability of having 2 defects: P(2) = P (uneven colour and damaged glaze) or P (uneven colour and distorted shape) or P (damaged glaze and distorted shape)
P(2) = (1/20 x 3/50) + (1/20 x 1/25) + (3/50 x 1/25) = 37/5000

Probability of having 3 defects: P(3) = P (uneven colour, damaged glaze and distorted shape)
P(3) =(1/20 x 3/50 x 1/25) = 3/25000

Therefore P (≥1 defects) = P(1) or P(2) or P(3) = 3/20 + 37/5000 + 3/25000 = 1969/12500 = 0.15752 = 15.8%


So my answer to 1 d.p is 15.8%, but it was marked incorrectly? Please can anyone see where I went wrong?



Thank you
 
A company producing glass vases is experiencing quality issues and has identified the following independent problems.

5% of the vases the colour is uneven.
6% of the vases the glaze is damaged.
4% of the vases have a distorted shape.

What is the overall probability of a vase having one or more of these defects?
Isn't "having one or more of these defects" the opposite of "having zero defects"?

What is the probability of zero defects?

Then, subtracting, what is the probability of having at least one defect? ;)
 
Hi, Thank toy for the reply.

The probability of having zero defects is 85% (100% - 6% - 5% - 4%).

I'm not totally sure what to do next though?

85% - (Probability of having 1 defect)?

85% - 15% = 70%?


I'm sorry, I don't really understand. :(
 
Well, let's think about what we have here. You (should) know that the odds of any event happening are always 1 (100%) minus the probability of the event not happening. It has also been established that, because "having one or more defects" is the opposite of "having zero defects," the only way for the event "having at least one defect" not to happen is for there to be zero defects. So, symbolically, if we adopt the notation P(D=x) to mean "having exactly x defects," then:

\(\displaystyle P(D \ge 1) = 1 - P(D=0)\)

Can you finish up from here?
 
Well, let's think about what we have here. You (should) know that the odds of any event happening are always 1 (100%) minus the probability of the event not happening. It has also been established that, because "having one or more defects" is the opposite of "having zero defects," the only way for the event "having at least one defect" not to happen is for there to be zero defects. So, symbolically, if we adopt the notation P(D=x) to mean "having exactly x defects," then:

\(\displaystyle P(D \ge 1) = 1 - P(D=0)\)

Can you finish up from here?

Hello, thank you for the reply too :)

So if \(\displaystyle P(D \ge 1) = 1 - P(D=0)\) and the probability of having no defects is 85% or 0.85, I would get \(\displaystyle P(D \ge 1) = 1 - 0.85\) \(\displaystyle = 0.15 =\) 15%?
 
Let's try this a different way.

Let p = the probability of bad color = 0.05.

Let q = the probability of good color = 1 - 0.05 = 0.95.

Let r = the probability of bad glaze = 0.06.

Let s = the probability of good glaze = 1 - 0.06 = 0.94.

Let t = the probability of bad shape = 0.04.

Let u = the probability of good shape = 1 - 0.04 = 0.94.

Are you ok so far?

Let a = the probability of good glaze given good color.

Can you determine this? Why?

Let b = the probability of good color and good glaze.

Is there a standard formula for this? If so, what is it?

Let c = the probability of good shape given good color and good glaze.

Let d = the probability of good color, glaze, and shape.

Let e = the probability of one or more defects 1 - d.

What are a, b, c, d, and e?

EDIT: Your fundamental difficulty here is this.

\(\displaystyle P(X\ or\ Y) = P(X) + P(Y) - P(X\ and\ Y).\)

You cannot add up probabilities as you have been doing unless events are mutually exclusive meaning that

\(\displaystyle P(X\ and\ Y) = 0.\)
 
Last edited:
Hi everyone, so I have tried this again and got:

Probability of a bowl of having bad color, P(Cb) = 5%
Probability of a bowl having a good color, P(Cg) = 100-5 = 95%

Probability of a bowl of having a bad glaze, P(Gb) = 6%
Probability of a bowl having a good glaze, P(Gg) = 100-6 = 94%

Probability of a bowl of having a bad shape, P(Sb) = 4%
Probability of a bowl having a good shape, P(Sg) = 100-4 = 96%

Probability of good glaze given good color, a = not applicable

Probability of good glaze and good color, b = 94% x 95% = 89.3%

Probability of good shape given good color and good glaze, c = not applicable

Probability of good color, glaze and shape, d = 94% x 95% x 96% = 85.728%

Probability of 1 or more defects, e = 1 – d = 85.728% = 0.14272 = 14.3%


So I got 14.3% - does that sound correct please?
 
Hi everyone, so I have tried this again and got:

Probability of a bowl of having bad color, P(Cb) = 5%
Probability of a bowl having a good color, P(Cg) = 100-5 = 95%

Probability of a bowl of having a bad glaze, P(Gb) = 6%
Probability of a bowl having a good glaze, P(Gg) = 100-6 = 94%

Probability of a bowl of having a bad shape, P(Sb) = 4%
Probability of a bowl having a good shape, P(Sg) = 100-4 = 96%

Probability of good glaze given good color, a = not applicable

Probability of good glaze and good color, b = 94% x 95% = 89.3%

Probability of good shape given good color and good glaze, c = not applicable

Probability of good color, glaze and shape, d = 94% x 95% x 96% = 85.728%

Probability of 1 or more defects, e = 1 – d = 85.728% = 0.14272 = 14.3%

So I got 14.3% - does that sound correct please?
I'm not sure what you're doing in much of the above, but you do appear to have arrived at the correct value.

In future, try working it the way that was suggested a few times previously in the thread:

. . . . .(probability of at least one problem) is

. . . . . . . .(prob. of any outcome) less (prob. of no problems)

This computes, as described earlier, as:

. . . . .1 - (0.95)(0.94)(0.96) = 0.85728

;)
 
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