Problematics | The mathematics of planning a family

Here's week 18 of Problematics.

Among the many memorable moments that have followed Argentina’s World Cup triumph, the images of Lionel Messi celebrating with his family are the standouts for me. Not only is the joy on the faces of Messi, Antonela Roccuzzo and their three sons infectious, but the images also bring to mind a mathematical question: How common, or how rare, is it for a couple to have three sons and no daughters?

Here's week 18 of Problematics.

When a family has three children, they can be born in any one of eight possible orders: BBB, BBG, BGB, BGG, GBB, GBG, GGB and GGG. So, the probability of BBB is ⅛. Alternatively, since a son is 50% probable (½) at each birth, the probability of three successive sons is ½ x ½ x ½ = ⅛.

Probabilities of sons and daughters being born are staple for puzzlers. I hope to bring you such puzzles from time to time, some of which will be my originals. But there are also some good ones which aren’t my own, and we cannot leave those out of a column that aims to cover every variety of puzzles in the long run.

The following has been adapted from a puzzle that I have found in more than one publication, in slightly different forms, but whose original creator I do not know.

#Puzzle 18.1 

In a state in India where the sex ratio has traditionally been skewed, the administration of one district sets out to correct the ratio within the geographical limits of its own jurisdiction. Simply put, the district planners hope to add more new daughters and fewer new sons, and so they devise these birth control rules. 

Clause 1.1: “Subject to Clause 1.2, a couple may have as many daughters as they want, as long as daughters are indeed born to them.” 

Clause 1.2: “If a son is born to a couple, that couple will not be legally allowed to have any more children.” 

The penalties for violation, which are detailed under another clause, are reasonably strict but not relevant to our puzzle. What does concern us is the reasoning behind these rules. As a district planner explains to a junior officer: 

“If a couple’s first child is a son, they will have no more children. On the other hand, if their first child is a daughter, they can have another child who may again be a daughter, and then have a third child who may be a daughter yet again. They can go on having children until they have their first son, at which stage they must stop. That way, no couple will have more than one son, but many couples will have a number of daughters. Overall, future births will include more daughters than sons, so the sex ratio is bound to improve.” 

We can agree with a couple of the assumptions that this theory is founded on: (i) at each birth, a couple will have either a son or a daughter; (ii) at any birth, a boy and a girl are equally probable. 

But, will the grand objective of ensuring more daughters than sons work? 

#Puzzle 18.2 

How about a physics puzzle? Load a boat with a couple of full dustbins and row out into the centre of a small pond. Dump the contents of both bins into the pond. Does the water level rise, fall or remain the same? 

 

Mailbox: Last week’s solvers 

Puzzle 17.1 

Hello Kabir, 

I noticed these puzzles for the first time and it felt really great to warm up our minds on a lazy Monday morning. The husbands and wives are: 

Jammu man and Odiya woman 

Kashmiri man and Naga woman 

Ladakhi man and Manipuri woman 

The Punjabi woman is single. 

— Abhiraj Malik, Delhi 

# Puzzle 17.2 

Hello Kabir, 

The number obtained after various permutations and combinations is 381654729. 

Hopefully this secret code opens various other puzzles. 

— Nipun Bamania, Mumbai 

Solved both puzzles: Abhiraj Malik (Delhi), Nitish Kumar, Varsha Jain (Mumbai), Yojit Manral (Faridabad), Kshitij Kumar (Delhi), Avneesh Kumar (Delhi), Shivika Gupta (Delhi), Jasvinder Singh (Nabha), Shashi Bahadur, Aman Chaudhary, Nipun Bamania (Mumbai), Dr R K Sharma (Delhi), Madhuri Patwardhan (Thane), Amardeep Singh (Meerut), Gopal Menon (Mumbai), Vaibhav Gautam (Delhi), Vinod Mahajan (Delhi), Naresh Dhillon (Gurgaon), Rahul Agarwal (Bay Area, California), Shishir Gupta (Indore), Rituparna Gupta (Indore), Ajay Ashok (Mumbai), Ravi Sondhi and Rudra Sondhi (Gurgaon), Roona Poddar Gupta 

Solved #Puzzle 17.1: Vinod Kumar Vij (Rohtak), Namrata Vala, Pallavi Kanchan, Kirthivasan S (Tamil Nadu), Yogender Narayan, Ankit Sud (Nadiad), Tanishi Goyal, Mahesh Mundhra (Indirapuram), Mrunal Patil, Ananthakrishnan Subramanian, Sanjay Gupta (Delhi), Pallavi, Rina Kumari (Noida Extension), Avanti Kashikar (Mumbai) 

Solved #Puzzle 17.2: Anu Kathpalia (Delhi), Anuradha V, Narendra Gala (Mumbai), Pragya Khurana (Ludhiana)

Problematics will be back next week. Please mail your replies to problematics@hindustantimes.com