Here is a question from a test I recently I just took. My teacher and I disagree on the answer.
"A printer printed a card with black ink on blue paper and an envelope with blue ink on blue paper. If the colors were chosen at random from a stock of 7 inks and 6 papers, what was the probobility of the given result?"
He thinks it is 1/21. I think it is 1/1764. He says that the reason that it is 1/21 is that there are two seperate events, but he cannot explain this any further, and I am not convinved.
Can anyone help me on this?
The total number of ways a paper/ink choice combination could come out is given by p*i. So, it's 42. You seem to know that already.
So, the chance of any one of those choices coming up is 1/42. You need to figure out the probability that two paper/ink choices will come out a certain way.
The choices made in one event to not affect the probability of any given choice coming up in a second event. So, the probability of the first event happening the way it did is 1/42, and the probability of the second happening the way it did is also 1/42.
I think that, to put them together, you multiply them. This comes out 1/1764, like you said. Your teacher appears to have simply multiplied by the number of events (2), giving 1/21. That would mean that, as you add more events, the likelihood of any given series of events coming out a certain way increases, which just makes no sense at all.
To see why, imagine flipping a coin. The probability that a certain face will come up is 1/2. If you want the same face to come up again, the chances are (1/2)(1/2) = 1/4. If you were to multiply by the number of events (2), you would get a probability of 1. This would mean that a coin, once flipped, *must* turn up the same face in all subsequent flips. This is clearly not the case, and if your teacher disagrees with you, ask him if he's ever flipped a coin before.
I mean, I could be wrong, of course. Anyone?
El que pega primero pega dos veces.
Your teacher is wrong.
"A printer printed a card with black ink on blue paper and an envelope with blue ink on blue paper."
If it was "or", then it would be 1/21.
Explain it like this:
The chance of event A happening is 1/42, and the change of event B happening is 1/42. Are you telling me that the chances of both of them happening is greater than the chance of just a single one? Doesn't that sound wrong?
"In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..."
you choose black ink for the card: 1/7;
you choose blue paper for the card: 1/6;
blue ink for the envelope: 1/7;
blue paper for the envelope: 1/7;
we have to multiply probabilities (black ink and blue paper and blue ink and blue paper) - we substitute and with multiplication:
so the answer is: 1/1764