Today, Alice is playing games-mistress with her friend Marcus. To win the game, Marcus must pick two marbles of the same colour.

At the start, there are 6 marbles placed inside a small bowl. They are identical to the touch but 3 are blue and 3 yellow. Two empty pouches lie next to the bowl. The player is then blindfolded, the marbles randomised, and he then picks 3 marbles and places them inside one pouch, with the other 3 marbles going into the other pouch. He is then asked to select one pouch and pick just one marble from that pouch and put it inside the other one. At that point, one pouch will have 2 marbles and the other 4 marbles and the blindfolds can be removed. Marcus cannot see inside each pouch but he can feel which one has 2 and which 4 marbles. Lastly, he selects 2 marbles from just one chosen pouch. If the two colours are the same, he wins, if they are different, he loses.

The question is whether it is better to select the pouch with 2 marbles or 4 marbles? Does it make a difference, and if so, what are the relative probabilities to win?

The game is then repeated, with Alice being the player and Marcus the games-master. The whole process described above is merely to ensure that the player is confident that the game is random and not manipulated by a potentially unscrupulous opponent!

There is a follow-up to this but I shall post it separately.

**IMPORTANT CHANGES TO THE PRIZE MATHS QUIZ!**

**YOU MAY NOW DISCUSS THIS PMQ PROBLEM IN THE COMMENTS BELOW AND GIVE YOUR ANSWERS AND SOLUTION METHOD.**

*I SHALL POST A GUIDED SOLUTION ON THE FOLLOWING FRIDAY AT THE SAME TIME AS THE NEW PMQ. AS THERE ARE OFTEN MULTIPLE WAYS OF ARRIVING AT THE SAME ANSWER, I SHALL ALSO GIVE CREDIT AND LINK TO THE BEST ANSWERS SUBMITTED IN THE COMMENTS.*

*IF YOU ARE A STUDENT INTERESTED IN THE PRIZE BELOW, THEN YOU MAY STILL SUBMIT YOUR ANSWER BY EMAIL. BUT, FROM NOW ON, ONLY ELIGIBLE STUDENTS, PLEASE.*

*I HOPE THIS WILL MAKE THE COMMENTS MORE VIBRANT!*

**You can receive these questions directly to your email box or read them in an RSS reader. Subscribe using the links on the right.**

**If you would like to DONATE PRIZES for the next PMQ, send me a message here.**

**How to Enter****IF YOU ARE A STUDENT, you may still send your complete solution by email to pmq31gauss@giftedmaths.com.**This email address shall be removed after the competition closes to avoid spam.

**This PMQ31 competition closes on WEDNESDAY 14 AUGUST at 23:59 GMT.**

**The Prize****The prizes for this PMQ are 3 free places in our Online Maths Club for ONE YEAR.**The very first correct solution will receive a prize plus two others randomly selected from all the other correct answers. The email time stamp shall determine the order of entries received. All winners can have their name posted and a link to their own online profile at their favourite social network or their own blog.

**Quick Rules****Look at the expanded rules on our PMQ page.**

Anybody can enter our Prize Maths Quiz; adults and students.

Use the official server clock in the right column to avoid late entries.

All emails and email addresses sent to us will be deleted after the winners have been processed.

DO NOT submit your entries to the comments section at the bottom of this post.

You CAN discuss it there but you must email us to enter the competition.

We adhere to COPPA guidelines regarding children's online safety and security.

Enjoy the challenge!

**Send us your solution! You can't win if you don't participate.**

Then tell your friends!