Solution
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To calculate the value of 9 factorial, denoted as
Here is the step-by-step calculation:
Here is the step-by-step calculation:
Here is the step-by-step calculation:
Now, let's perform the multiplication step by step:
Therefore, the value of
Now, let's perform the multiplication step by step:
Therefore, the value of
Exercise 11 - Parts d-h- How to calculate the probability of four couples sitting in the eight seats?
The Binomial Theorem
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Exercise 11 - Parts d-h- How to calculate the probability of four couples sitting in the eight seats?
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In this section, we're asked what's the probability
that each of the Smith has a window seat?
We can have the male Smith here in the female Smith here,
or the female Smith here,
and male Smith here.
We really don't care what happens to the other guys.
Let's just get started here.
We have here in the denominator again,
8 factorial, which is the size of the sample space.
Now, we have here 2 people that are the Smith and we have here 2 seats.
In how many ways can I order 2 people in the 2 seats.
That's 2 factorial, which is 2.
Either the male Smith is here in the female Smith is here,
or the other way round where the female Smith is here and the male Smith is here.
That's 2 factorial.
Now we have 6 other people that sit in the 6 different seats.
We have 6 factorial ways of seeding the 6 people in the 6 remaining seats.
That's the probability and that equals to 0.0357.
In this section, we're asked what's the probability that each of
the Smiths seat where 1 faces forward and the other face is backwards.
For example, if the male sits here,
the female can sit in any 1 of the 4 seats here, for example here.
Or if the female sits here, for example,
well the male can sit in any 1 of the 4 seats here, let's say here.
Let's just build the probability here.
We have that. Now, what is that equal to?
In the denominator again,
we have the size of the sample space,
which is 8 factorial.
Let's calculate the numerator.
Let's take 1 of the Smith,
so I'm choosing 1 out of the 2.
There'll be 2 over 1.
Then let's say I put him in 1 of the seats facing forward.
We said that this was facing forward,
this was facing backwards.
I have to choose 1 out of the 4 here,
so that'll be 4 over 1.
Now the other Smith will have to choose 1 out of the 4 facing the other way,
so that'll be 4 over 1.
Now, I still have 6 remaining people to be seated in 6 remaining seats.
That has to be times 6 factorial,
that equals to 0.5714.
Let's just review again.
I took 1 of the Smith, right.
I chose 1 out of the 2 people and I placed them in 1 of the 4 seats.
For example, facing forward.
The other Smith while I chose 1 of the 4 seats facing the other direction,
and then I place the 6 remaining people in the 6 remaining seats,
and I divided that by 8 factorial.
This then would be the probability.
In this section, we're asked,
what's the probability that the Smiths sit facing each other or sit opposite 1 another.
For example, if the male sits here then the female has to sit here,
or if the female sits here,
then the male has to sit here.
Let's get started. Again,
in the denominator,
we have 8 factorial.
Now, let's choose 1 of the Smith.
I choose out of the 2 Smiths, I choose 1.
Now, I have to choose 1 of the 4 seats that that person will sit in,
so that'll be 4 over 1.
Now, the other Smith doesn't really have that much options,
so we have to multiply this by 1. Why is that?
Because again, if 1 Smith sits here,
then there's really no options.
The other Smith has to sit opposite the partner.
Again, we still have the 6 remaining people in the 6 remaining seats,
so that'll be 6 factorial.
Now, that comes out to 0.1429.
In this section, we're asked what's the probability that all the men
sit facing forward and all the woman's sit facing backwards.
Again, this would be the direction of travel.
These are the seats facing forward and these are the seats facing backwards.
We want all the men to be here and all the woman to be here.
Let's calculate this probability.
In the denominator, again,
we have 8 factorial.
Now, how many seats do I have here?
I have 4 seats facing forward.
How many men do I have here? Well, 4 men.
So that has to be 4 factorial.
The number of combinations of seating for men in the 4 seats.
I have 4 remaining seats facing backwards,
and I have 4 women that need to be seated in these 4 seats.
How many combinations do I have here?
Well, 4 factorial.
This comes out to 0.01429.
In this section, we're asked what's
the probability that each couple sit facing each other?
That means that the first couple would sit here,
the second couple here,
the third couple would sit here,
and the fourth couple would sit here.
What's this probability? First of all,
in the denominator we have the size of the sample space, That's 8 factorial.
Now here I'm looking at couples.
I have basically 4 objects.
I don't have 8 people,
am not looking at each individual person,
but I am looking at each couple.
I have 4 couples or 4 objects.
Now I have here then 4 places.
I have this place right here.
I have this place right here,
I have this place right here,
and I have this place right here.
How many ways can I order the 4 couples in each 1 of the places?
Well, that's 4 factorial.
Now, I need to deal with the internal ordering.
It can have the male and the female here, or vice versa,
that the female would be here and the male would be here enough for 1 couple,
there'll be 2 factorial.
Now, I need to do this 4 times,
so I need to take the 2 factorial and raise it to the power of 4.
Now, all of that comes out to 0.0095.
That's the probability that each couple would sit facing each other.
This video explains the probability of seating arrangements for 8 people, including two Smiths, in an airplane. The probability of each of the Smiths having a window seat, one Smith facing forward and the other facing backward, the Smiths sitting opposite each other, and all the men facing forward and all the women facing backward are all calculated. The probability of each seating arrangement is 0.0357, 0.5714, 0.1429, and 0.01429 respectively.
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