QR probability question

[super112]

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So I wrote this problem down awhile ago so I'm not sure if I copied the answer down incorrectly..


If a restaurant has 5 types of pasta and 4 sauces, what are odds of getting pasta type 1, sauce 1, or BOTH pasta type 1 and sauce type 1?


I did 1/5 + 1/4 + (1/5 * 1/4), but I previously wrote down 1/5 + 1/4 - (1/5 * 1/4). Help please?

 

[FeralisExtremum]

The latter calculation 1/5 + 1/4 - (1/5 * 1/4) is correct. You have to subtract the "both" event because it is technically double counted when you add the two separate probabilities. In general, just try to remember that for independents events, p(A or B) = p(A) + p(B) - p(A and B)

 

[DAT Destroyer]

The question is not asking for the PROBABILITY of getting pasta type 1, sauce 1 or both. It's asking for the ODDS.

The odds of an event = (chances for the event)/(chances against the event)
Chances for are: (1/5)+(1/4)-(1/5*1/4)=8/20=2/5
Chances against= 1-2/5=3/5
Therefore the odds are: (2/5)/(3/5)=2/3

Hope this helps.

 

[super112]

I'm glad you mentioned the thing about the odds! I did not know that and I'm betting I was trying to write the problem shorthand because the answer is 2/5 as you said.

in regards to, p(A or B) = p(A) + p(B) - p(A and B). Wouldn't that only be if they asked chance of pasta 1 OR sauce 1? Instead they ask: pasta 1 OR sauce 1 OR both pasta 1 and sauce 1.

Am I mistaken to interpret it like that?

 

[FeralisExtremum]

in regards to, p(A or B) = p(A) + p(B) - p(A and B). Wouldn't that only be if they asked chance of pasta 1 OR sauce 1? Instead they ask: pasta 1 OR sauce 1 OR both pasta 1 and sauce 1.

Am I mistaken to interpret it like that?

Things can get confusing here, but when asked for A or B (two independent events), the probability of that outcome includes (by default) the possibility of A and B, even if you're not specifically asked for it. So whether I asked you for the probability of A or B, or I asked for the probability of A or B or both - the calculation is the same. The question will usually specify if it does not want you to include the possibility of both events occurring, phrased like "What is the probability of A or B but not both?" - then you would solve like so:

p(A or B but not both) = p(A) + p(B) - [2 * p(A and B)]

I wouldn't worry about it too much because the question usually specifies either way if you should include or exclude the "both" outcome.

 

[DAT Destroyer]

Yes that is correct.