Why the Sum of all possible outcomes does not equal to one, in this case?

The question is an extension from an example (click this--> Introduction to Probability and Its Applications by Richard Scheaffer, Linda Young. The link points to the exact question/solution.

Edit:-

Five cans of paint (numbered 1 through 5) were delivered to a professional painter. Unknown to her, some of the cans (1 and 2) are satin ﬁnish and the remaining cans (3, 4, and 5) are glossy ﬁnish. Suppose she selects two cans at random for a particular job. Let A denote the event that the painter selects the two cans of satin-ﬁnish paint, and let B denote the event that the two cans have different ﬁnishes (one of satin and one of glossy). Find P(A) and P(B).

It gives a solution of:

P(A) = Satin, Satin = 2/20 = 0.1

P(B) = Mixed finishes = 12/20 = 0.6

My question is what are the other possibilities 🤔 I think Glossy and Glossy finishes only.

Okay that's {3,3 3,4 ... 5,5} = Nine such tuples... Out of 20 giving us 9/20.

2/20 + 9/20+ 12/20 > 1

I am confused why this does not Equal to one, since all the possibilities are exhausted.

Thank you.

• Can you please post the full text of the problem? Though, I'm sure the answer will be "you're calculating ____ wrong" as it's an axiom that the sum of probabilities is 1 – Stella Biderman Feb 5 '16 at 4:04
• You should write down the entire problem. I do not even have access to a preview of the text, many others may not as well. – Cameron Williams Feb 5 '16 at 4:27
• Your link does not go to the question, it just goes to the book cover. The language after My question is incoherent, We haven't heard of Glossy before, the set is unexplained and does not seem to consist of tuples. – Ross Millikan Feb 5 '16 at 4:30
• Cameron and Ross, I have attached the question now. Would appreciate your input thanks. – weallneedmoresunshine Feb 5 '16 at 4:32
• You are correct that it is glossy/glossy only. You made an error in just counting the possible outcomes. The outcomes are not independent. If you select one glossy paint can, the probability of selecting another is lessened. The probability of one is $3/5$ and the probability of a second is $2/4$ so you get an overall probability of $6/20$ which gives a total probability of $1$. – Cameron Williams Feb 5 '16 at 4:36