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Conditional Probability sequential exams.

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G is planning to take a sequence of exams which are reliant on the previous exam to be passed in order to move onto the next. G calculates the probability of passing the first exam as 0.9, passing the second exam on the condition she passes the first exam as 0.8 and passes the third exam on the condition she passes the first two as 0.7. Whats the probability she passes all 3 exams? Solution: let A be the event she passes first exam. B the second and C the third. P(ABC)=P(A)P(B|A)P(C|BA)=9/10*8/10*7/10 = 63/125 Part B: given she did not pass all three exams what is the conditional probability she failed the second exam? i know that P(ABC) is passing all exams, so logically taking the compliment wouild be $$P(ABC)^{c}=1-P(ABC)=P(A^{c} \cup B^{c} \cup C^{c})$$ so that makes me consider the $$P(B^{c}|(ABC)^{c})$$ which in all honesty confuses me as this leads to

$$P(B^{c} \cap (ABC)^{c})= P((B^{c} \cap A^{c}) \cup (B^{c} \cap B^{c}) \cup (B^{c} \cap A^{c}))$$

as these events are not indepdent of each other.

any suggestions?

Guest Jul 11, 2017
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The probabiltity that she did not pass the second exam.

In order to take the second exam, she must have passes the first.

So, in order to pass the first exam and to fail the second:  0.9 x 0.2  =  0.18.

After failing the second exam, she is not allowed to take the third exam.

I believe that the answer is 0.18.

geno3141  Jul 12, 2017
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Geno, you have the hippo but you are missing the potamus.

(Using subscripted variables makes it easier to present and keep track of the relations.)

$$\small \text {Let A1 = {pass exam 1}, A2 = {pass exam 2}, A3 = {pass exam 3}, and A = {passes all three exams}. }\\$$

$$\text{Because }\ A^c_2\subset A^c, \; A^c_2\cap Ac=A^c_2 \\ \mathbb{P}(A^c_2) = \mathbb{P}(A^c_2|A^c_1)\; \cdot\; \mathbb{P}(A^c_1 +\mathbb{P}(A^c_2|A_1) \; \cdot\; \mathbb{P}(A_1) \leftarrow \small \text{Apply Total Probability Formula}\\ = (0 \cdot 0.1) + (0.2 \cdot 0.9) = 0.18 \\ \text {Then }\\ \mathbb{P}(A^c_2|A_c) = \dfrac {\mathbb{P}(A^c_2 \cap A^c)} {\mathbb{P}(A^c)}\\ = \dfrac {0.18}{0.496} = 0.363$$

GA

GingerAle  Jul 12, 2017
edited by GingerAle  Jul 12, 2017
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Rhinoceros !!!!!!!

noun, plural rhinoceroses (especially collectively) rhinoceros.

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any of several large, thick-skinned, perissodactyl mammals of the family Rhinocerotidae, of Africa and India, having one or two upright horns on the snout: all rhinoceroses are endangered.

Guest Jul 12, 2017
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Thanks guest,

Yep, Ginger sure got that one wrong :)

Gino is definitely affiliated with rhinos, not hippos :))

Melody  Jul 12, 2017
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Actually, I didn’t get it wrong.  My comment has nothing to do with Geno’s avatar. It is simply a metaphor I use when something is about half completed.

“I’ve completed the hippo and I’m working on the potamus,” is my typical reply when a patron asks about the progress on a commissioned portrait. One time I responded this way to a patron who asked about my progress on his wife’s life-size portrait. He was rather offended and I had to explain I wasn’t referring to his wife as hippopotamus.

Because of my troll-chimp nature, I added that she is rather hippy and does have the largest potamus I’ve seen in a long time.  He was awkwardly quiet for a long time, but he didn’t cancel the commission until I said that I bought the economical 37-liter vats of oils.  It’s a pity that some people just do not have a sense of humor.

Geno’s avatar does look somewhat like a hippopotamus when displayed in thumbnail size.  I kinda, sorta figured a blarney bag would come along and point out the obvious error of my ways.  He did. And he says, “All rhinoceroses are endangered.” Why, I had no idea!!! That’s a good thing to know. I’ll cut back on my use of rhinoceros horn. I wouldn’t want to be a contributor to the extinction of rhinocerosesjust blarney bags.

GingerAle  Jul 12, 2017
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Hi Ginger,

I think your metaphors could be customized but not to worry,

You cracked me up with:

"I wasn’t referring to his wife as hippopotamus....

Because of my troll-chimp nature, I added that she is rather hippy"

Thanks for the early morning humour, it is a good way to start the day.