# Math Help - Deck of 52 cards

1. ## Deck of 52 cards

I need help setting this problem up. Thanks.

A Deck of 52 cards is shuffled. The cards are dealt face up one at a time until an ace appears.

1) What is the probability that the first ace appears on the third car?
2) Show that the probability of getting the first ace on or before the ninth card i greater than 50%.

2. ## Re: Deck of 52 cards

Originally Posted by Eminem
A Deck of 52 cards is shuffled. The cards are dealt face up one at a time until an ace appears.
2) Show that the probability of getting the first ace on or before the ninth card i greater than 50%.
Here is the calculation.

3. ## Re: Deck of 52 cards

This belongs in Probability and Statistics, not pre-calc.

It is very hard to help you because you have not shown any work so we don't know where you are stuck.

Can you compute the probability that the first two cards are not aces? Given that the first two cards are not aces, what is the probability that the third is an ace?

Now can you put those facts together to answer your first question?

4. ## Re: Deck of 52 cards

Originally Posted by JeffM
This belongs in Probability and Statistics, not pre-calc.
This is almost impossible to stop. I gave up several years ago.

Originally Posted by JeffM
Can you compute the probability that the first two cards are not aces? Given that the first two cards are not aces, what is the probability that the third is an ace?
I do not think that this has any use in answering the second question.
Had the question been put as: "What is the probability that the first ace appears within the first three deals?"

Originally Posted by JeffM
Now can you put those facts together to answer your first question?
The answer that I gave is what is often called "a back-of-the-book solution".
The idea being to present the learner with the complete solution and then demand an explanation from the learner.

But hay what do I know, I only taught probability theory at a university for over thirty years?

5. ## Re: Deck of 52 cards

Originally Posted by Plato
This is almost impossible to stop. I gave up several years ago.

My turn then

I do not think that this has any use in answering the second question.
Had the question been put as: "What is the probability that the first ace appears within the first three deals?"

Agreed, but the student still must cope with the first question as posed, or have I misunderstood your point.

The answer that I gave is what is often called "a back-of-the-book solution".
The idea being to present the learner with the complete solution and then demand an explanation from the learner.

But hay what do I know, I only taught probability theory at a university for over thirty years?
I am being curious rather than argumentative. If the best approach here is to give an answer and then ask the student to explain why it is valid, then why not give a simple answer such as

$1 - \left\{ \dbinom{48}{9} \div \dbinom{52}{9}\right\} \approx 0.54415.$

6. ## Re: Deck of 52 cards

Originally Posted by JeffM
I am being curious rather than argumentative. If the best approach here is to give an answer and then ask the student to explain why it is valid, then why not give a simple answer such as
$1 - \left\{ \dbinom{48}{9} \div \dbinom{52}{9}\right\} \approx 0.54415.$
You could certainly do it that way. I should think that is a multi-part question one wants the parts to lead somewhere.

My objection was in this two part question, I don't how that is the case.

7. ## Re: Deck of 52 cards

Originally Posted by Plato
You could certainly do it that way. I should think that is a multi-part question one wants the parts to lead somewhere.

My objection was in this two part question, I don't how that is the case.
Ah Ha. I think I get your point now. Sorry if I was being dense.

You believe the problem is designed to focus on $P(A_1\ or\ A_2\ or\ ...\ A_9)$, where the various A's are mutually exclusive. So my initial answer (or my question in my second post) did not sharpen that focus. Indeed my question in the second post never even considered the possibility that the two parts of the problem were intended to be closely linked.

All I can say in defense of my original post is that it's only purpose was to get the student to find the probability of $A_3$, which was a necessary part of the exercise. Obviously, it did not encourage the generalization the problem creator intended, but it did not discourage it either. When students show no work, it's hard to figure out what is the immediate help required. Also I am not used to thinking about how to further the design of a problem; thanks for giving me something new to think about going forward.