Probability - ways in which its possible

• JessBrown
In summary, there are 216 different ways in which two people can read four books at the same time, each person reading one book at a time, where the books cannot be read in their original order. This is known as the derangements problem and can be solved by finding all possible combinations and then eliminating those where the books are in their original order.
JessBrown
So there are four books to be read by 2 people, same time to read a book each, and the question is how many different ways can they read the four books?

So I firstly took the approach that you have:
Person 1: Spot1 Spot2 Spot3 Spot4
Person 2: Spot1 Spot2 Spot3 Spot4
And then filling in how many book options there are in each spot.

So Person 1 Spot1 has 4 options.
Whereas Person2 Spot1 then has only 3 options (as one book is being read).
Therefore Person 1 Spot 2 then has 3 options left (Person1 has read one book already).
My biggest issue is in Person 2 Spot 2, I think there are 3 options, but not sure! (As Person2 has read one book, so has three remaining??) (or only 2 options?)

Then Person 1 Spot 3 has 2 opitons, Person 1 Spot4 has 1.
Person 2 Spot 3 I think has 2 remaining (or 1) and Person 2 Spot 4 has 1

Sooooo I think there are
Person 1: 4 3 2 1
Person 2: 3 3 2 1
Which results in 432 different ways in which to read the books... is this right though cos I have a feeling it may be
Person 1: 4 3 2 1
Person 2: 3 2 1 1
Which means there are only 144 options?

Thanks!

This one's a nice twist on the old derangements problem. The correct answer is somewhere between the two numbers you guessed and a bit trickier to compute.

Ok, so you found that person 1 can read the books in 24 different ways. Suppose person 1 chose any order and then relabel the books 1,2,3,4 in that order. Person 2 now must choose an ordering so that 1 isn't 1st, 2 isn't 2nd etc. Now there are only 24 combinations to check which can be done by brute-force, i.e. write them all out and find the valid ones (it will be an odd number).

Sooooo I wrote them all down... and the answer I got was still 216, as there are nine combinations for each of the first row's orders... so 24 ways to order the first row, means 24x9 ways to order in total... the first way was so much quicker though!

eg.
1234

2143
2341
2413
3142
3412
3421
4321
4312
4123

gives nine ways for the combo of 1234 on top row, therefore there are 24 different ways for the top row, 216 in total!

anyone want to check this for me? I am fairly sure now...

I would approach this problem using the fundamental principle of counting. This principle states that if one event can occur in m ways and a second event can occur in n ways, then the two events together can occur in m x n ways. In this case, the two events are the choice of books for Person 1 and Person 2.

So, for Person 1, there are 4 books to choose from for the first spot, 3 books for the second spot, 2 books for the third spot, and 1 book for the last spot. This gives us 4 x 3 x 2 x 1 = 24 options for Person 1.

For Person 2, there are also 4 books to choose from for the first spot, but then only 3 books for the second spot (since one book has already been chosen by Person 1), 2 books for the third spot, and 1 book for the last spot. This gives us 4 x 3 x 2 x 1 = 24 options for Person 2.

To find the total number of ways in which the books can be read, we multiply the options for Person 1 and Person 2 together: 24 x 24 = 576 possible ways.

So, the correct answer would be 576 different ways in which the four books can be read by two people at the same time. It is important to note that this assumes that each person can only read one book at a time and that the order in which they read the books does not matter. If these assumptions are not true, then the number of possible ways would be different.

1. What is the definition of probability?

Probability is the measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents impossibility and 1 represents certainty.

2. What are the different ways in which probability can be calculated?

The two main ways of calculating probability are the classical method and the empirical method. The classical method uses theoretical probabilities based on equally likely outcomes, while the empirical method uses observed frequencies of events to determine probabilities.

3. What is the difference between independent and dependent events in probability?

Independent events are events that do not affect the probability of each other occurring, while dependent events are events that do affect the probability of each other occurring. In other words, the outcome of one event does not influence the outcome of the other in independent events, but it does in dependent events.

4. How can probability be represented visually?

Probability can be represented visually through Venn diagrams, tree diagrams, and probability tables. These tools help to visualize the different possible outcomes and their corresponding probabilities.

5. What is the importance of understanding probability in real-life situations?

Understanding probability is crucial in making informed decisions and predictions in real-life situations. It allows us to assess risk, make strategic decisions, and interpret data accurately. Probability is also essential in various fields such as finance, medicine, and science.

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