We say that a quadrilateral is a bow-tie if two of the sides cross each other. An example is shown below.

Seven different points are chosen on a circle. We draw all chords that connect two of these points. Four of these 21 chords are selected at random. What is the probability that the four chords form a bow-tie quadrilateral?

This is a counting and probability question. I’m not sure how to handle it and I don’t even know how to begin. If someone could help me with this, I’d really appreciate it! 🥰😅

ArchedScythe Nov 11, 2019

#7**+1 **

Sure! I’ll be happy to share my thoughts about this problem! (Sorry for not doing it initially. 😁)

Here's a pretty concise explanation:

We know that a quadrilateral needs to have four vertices (or points on the circle). There are always two ways to link the cross — horizontally or vertically. Using my limited knowledge of combinations, we know that choosing four points out of seven equals 35. Multiplying the two ways to connect those lines (again, horizontally and vertically) makes 35*2 = 70 "bow-tie quadrilaterals" that can be formed on the circle using four points. There are 5985 ways four chords can be chosen out of twenty-five chords because C(25,4) equals 5985, so the probability is 70/5985... and then we just need to simplify that fraction. 😉

I hope that was an acceptable summary? I just want to thank CalculatorUser his/her guidance and I apologize to Melody for not initially putting my explanation! (I don’t know if that makes sense, so if anybody wants a more detailed summary, just let me know. 🥰)

ArchedScythe Nov 17, 2019

#1

#3**+1 **

Hi, I’m not really sure how you got the answer. Can you further explain it and/or your techniques to get the answer?

ArchedScythe
Nov 11, 2019

#4**+1 **

First find the total number of ways four cords can be chosen out of 21 cords.

Can you do that?

Then count the number of bow tie cords there are.

We do this by listing the TYPES of bow tie cords.

Then counting for each cord type, after this, we add them up all together.

We take the result and divide it by the total number of ways four cords can be chosen out of 21 cords.

CalculatorUser Nov 12, 2019

#6**0 **

Good.

But why not share your thoughts/solution with other people who have shown interest in your question. ?

It would be good if you did

Melody
Nov 13, 2019

#7**+1 **

Best Answer

Sure! I’ll be happy to share my thoughts about this problem! (Sorry for not doing it initially. 😁)

Here's a pretty concise explanation:

We know that a quadrilateral needs to have four vertices (or points on the circle). There are always two ways to link the cross — horizontally or vertically. Using my limited knowledge of combinations, we know that choosing four points out of seven equals 35. Multiplying the two ways to connect those lines (again, horizontally and vertically) makes 35*2 = 70 "bow-tie quadrilaterals" that can be formed on the circle using four points. There are 5985 ways four chords can be chosen out of twenty-five chords because C(25,4) equals 5985, so the probability is 70/5985... and then we just need to simplify that fraction. 😉

I hope that was an acceptable summary? I just want to thank CalculatorUser his/her guidance and I apologize to Melody for not initially putting my explanation! (I don’t know if that makes sense, so if anybody wants a more detailed summary, just let me know. 🥰)

ArchedScythe
Nov 17, 2019

#8**+1 **

**Thanks ArchedScythe,**

I have learned from you.

I just want to write your logic using different words. (note : for this question rotations are NOT the same)

There are 7 points on the circle.

Every chord must join 2 points so there there is a total of 7C2=21 possible chords

So

the number of ways to choose ANY 4 chords is 21C4 = 5985

That is the sample space.

NOW

To make a bow tie we must choose 4 points.

There are 7C4 = 35 ways to chose four points.

Now how many ways are there to join these 4 points into a bow tie using just 4 chords.

If you draw it on a peice of paper you will see that there are only 2 ways.

So

There are 70 ways a bow tie can be created.

P( the 4 chords create a bowtie) = 70 / 5985

Just like ArchedScythe said.

Melody Nov 17, 2019