Question

Pat picked a card from a standard deck, looked at it, and then put it back. He then picked a second card. What is the probability that Pat picked a diamond or a jack on either pick?

sixteen over fifty two

one over fifty two

thirteen over fifty two

seventeen over fifty two

Answer 1

what is the probability he got in on the first pick?

Answer 2

If he picked a card then replaced it into the deck, you have the same deck again in the 2nd pick.

So figure out the probability of picking a diamond, \(P(D)\),
the probability of picking a jack, \(P(J)\)

Then, the probability of picking a diamond OR jack (in one single pick), then \(P(D \text{ or } J ) = P(D \cup J) = P(D) + P(J) - P(D \cap J)\)

Now to get the probability of "diamond or jack" on either turn, means that you could get:
"Diamond or Jack" on pick 1, OR "diamond or Jack" on pick 2.

So:
turn 1: the probability is \(P(D \cup J)\)
turn 2: the probability is \(P(D \cup J)\)
The overall probability is then:
\(P(D \text{ or } J \text{ on either turn})=P(D \cup J)+P(D \cup J)\).. there is no intersection to worry about for this one because the 1st pick and 2nd pick are mutually exclusive

Answer 3

If you want to be explicit about the last statement I wrote^^, well you could theoretically write it in a more cumbersome way, following the "regular" rule for unions as:
\( P\left((D \text{ or } J)_{\text{turn 1 } }\cup (D \text{ or } J)_{\text{turn 2 } }\right)\\=P(D \cup J)_{\text{turn 1 } }+P(D \cup J)_{\text{turn 2 } }-P\left((D \text{ or } J)_{\text{turn 1 } }\cap (D \text{ or } J)_{\text{turn 2 } }\right)\)

But as I said, the intersection of the event \((D \text{ or } J)_{\text{turn 1 } }\cap (D \text{ or } J)_{\text{turn 2 } }\) is 0 since both turns cannot occur simultaneously.

Answer 4

ok soo im thinking its A

Answer 5

Ah hm yes I think you are right. According to the choices, adding P(D or J) + P(D or J) doesn't work. I guess I misinterpreted "either" as being "inclusive or", when I think it should rather mean "exclusive or", so the probability should just be P(D or J) which is the same as answer A