You have a 4-card deck containing a king, a queen, a jack, and a 10. You pick a card, keep it, and then pick another card. What is the probability that you pick exactly 1 face card? (Face cards are jacks, queens, and kings.)

- zarkam21

- jamiebookeater

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- anonymous

There are two ways to accomplish this: choose a face card first and then the 10, or choose the 10 first and then a face card. You need to calculate both of these probabilities and add them together.

- anonymous

So, let's begin. From the 4 cards, what is the probability that a face card will be chosen?

- zarkam21

3/4

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- anonymous

Terrific. You keep that card so now there are only 3 cards. What is the probability that you will then choose the 10?

- zarkam21

1/4

- anonymous

Not quite. Remember, you kept the first card so, for the second draw, there are only 3 cards? What do you think?

- zarkam21

2/4

- zarkam21

2/3

- anonymous

Nope. As an example, let's say you drew the queen. Now, what's left in the pack of cards? It's the king, the jack, and the 10. So, what is the probability you draw the 10 from this smaller pack containing only the king, jack, and 10?

- zarkam21

1/3

- anonymous

Right.

- anonymous

3/4

- anonymous

So, drawing a face card first had a probability of 1/4 and then drawing the 10 had a probability of 1/3. The probability of these two events happening is the product of these individual probabilities.\[P_{\text{facecard then 10}} = P_{\text{facecard}} \times P_{\text{10}}\]Can you calculate this?

- anonymous

* 3/4 sorry

- zarkam21

3/12 which is simplified to 1/4

- anonymous

Excellent. Now, we have to consider the other possibility.

- zarkam21

Okay

- anonymous

From the pack of 4 cards, what is the probability of choosing the 10 first?

- zarkam21

1/4

- anonymous

Great. Now, there will be only three cards left: King, queen, jack. What's the probability of choosing a face card from these three?

- zarkam21

3/3

- anonymous

Terrific. So the probability of these events happening is the product of the two probabilities, just like before. What do you get?

- zarkam21

same thing :]

- zarkam21

3/12 reduced to 1/4

- anonymous

That's right.

- zarkam21

thank you !!

- anonymous

Not done yet.

- zarkam21

oh okay

- anonymous

You've determined that the probability of face card then 10 is 1/4. You've determined that the probability of choosing 10 then face card is also 1/4. These are the only two ways the question can be satisfied. To get the final answer, you need to add these two probabilities together. What do you get?

- zarkam21

2/4 which is 1/2

- anonymous

Good job. Well done.

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