MHB Probability: Replacement & Without Replacement

[mathdad]

Can someone please explain the probability of replacement and without replacement with one simple example for each?

 

[MarkFL]

Can someone please explain the probability of replacement and without replacement with one simple example for each?

Suppose we have a standard deck of 52 cards, and we wish to compute the probability that when randomly selecting 4 cards from the deck, all 4 will be aces.

With replacement, we will always have 4 ways to draw an ace out of 52 possible draws, hence:

$$P(X)=\left(\frac{4}{52}\right)\left(\frac{4}{52}\right)\left(\frac{4}{52}\right)\left(\frac{4}{52}\right)=\frac{1}{13^4}$$

Without replacement, with each successive draw, we will have one less way to draw an ace and one less possible draws, hence:

$$P(X)=\left(\frac{4}{52}\right)\left(\frac{3}{51}\right)\left(\frac{2}{50}\right)\left(\frac{1}{49}\right)=\frac{1}{13\cdot17\cdot25\cdot49}$$

 

[mathdad]

The ideas are not easy to grasp.

 

[MarkFL]

The ideas are not easy to grasp.

Probability takes some practice, and even then it can be very difficult in certain scenarios. I find it to be one of the most challenging areas of math in which I have some kind of experience. :D

Luckily, I have grown my beard long so that I can stroke and tug on it when working probability problems. (Rofl)

 

[mathdad]

Probability word problems are tricky and require lots of practice. Wish I had more time on my hands.

 

[MarkFL]

Probability word problems are tricky and require lots of practice. Wish I had more time on my hands.

You would likely be better off getting your hands on a textbook dedicated to probability to learn the development of the elementary techniques. But, you are fortunate, there are several folks here who are quite knowledgeable in this topic and who can provide guidance if you delve into this area of study. (Yes)

 

[mathdad]

You would likely be better off getting your hands on a textbook dedicated to probability to learn the development of the elementary techniques. But, you are fortunate, there are several folks here who are quite knowledgeable in this topic and who can provide guidance if you delve into this area of study. (Yes)

Trust me, I will dive completely into a probability textbook in the coming months. Can you recommend a good probability textbook from amazon.com? Usually, it is not just probability alone. Most of the time, textbooks in this area of math include statistics.

 

[MarkFL]

Trust me, I will dive completely into a probability textbook in the coming months. Can you recommend a good probability textbook from amazon.com? Usually, it is not just probability alone. Most of the time, textbooks in this area of math include statistics.

No, I don't have a good recommendation...I would try your local library first. :D

 

[mathdad]

No, I don't have a good recommendation...I would try your local library first. :D

What about asking someone in this forum?

 

[MarkFL]

What about asking someone in this forum?

Those who are both interested in probability and who may have a recommendation will likely read this thread. While I know several people here who know a great deal about probability, I don't want to name any names because that would place undue pressure on those members. It is best to let people volunteer information free from being "called out." :D

The textbook I used as a student was "Introductory Statistics, 3rd Edition" by Weiss/Hasset, ISBN 0-201-17833-8, Copyright 1991 by Addison-Wesley Publishing Co., Inc.

 

[mathdad]

Those who are both interested in probability and who may have a recommendation will likely read this thread. While I know several people here who know a great deal about probability, I don't want to name any names because that would place undue pressure on those members. It is best to let people volunteer information free from being "called out." :D

The textbook I used as a student was "Introductory Statistics, 3rd Edition" by Weiss/Hasset, ISBN 0-201-17833-8, Copyright 1991 by Addison-Wesley Publishing Co., Inc.

Since I am not going to dive deep into probability right now, I will search through amazon when the time is right. I want to continue posting from the David Cohen precalculus textbook, which has over 500 questions. Good night.

 

[MarkFL]

Since I am not going to dive deep into probability right now, I will search through amazon when the time is right. I want to continue posting from the David Cohen precalculus textbook, which has over 500 questions. Good night.

Cohen might be a good source for problems, however, for development of the theory, there are likely much better texts dedicated to the topic...that's all I'm saying here. And you might find good texts at your local library, that you can read, gain a better understanding, then return, without putting out any cash. :D

 

[mathdad]

Cohen might be a good source for problems, however, for development of the theory, there are likely much better texts dedicated to the topic...that's all I'm saying here. And you might find good texts at your local library, that you can read, gain a better understanding, then return, without putting out any cash. :D

You are not the only person to tell me this fact. Based on my questions/solution steps, they feel that the best choice for me is the MATH FOR DUMMIES series, which I actually enjoy.