ERJ Brainteaser - May

Question 4: Sheep and goats

There are 12 sheep and 8 goats in a field. The farmer selects three animals, at random, to bring to market. What is the probability that the farmer chooses at least two sheep?

Answer: Some really neat ways of getting to the answer 65.6%. Very well done to: Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Bharat B Sharma, technical director, TWC Group- Rajsha Chemicals Pvt. Ltd & Techno Waxchem Pvt. Ltd, Vadodara, India; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; and everyone else who had a go.

SOLUTIONS

Amparo Botella

There are 12 sheep and 8 goats, so 20 animals total.

The farmer selects 3 animals at random.

We need to find the probability that at least 2 of the 3 animals are sheep.

We use combinations:

20*19*18

3*2*1       = 1140

1: Exactly 2 sheep and 1 goat

Choose 2 sheep from 12: 12*11/2 = 66

Choose 1 goat from 8: 8/1= 8

So, number of ways = 66*8=528

2: Exactly 3 sheep

Choose 3 sheep from 12: 12*11*10/3*2=220

Total favorable outcomes

528+220=748

So:

P (at least 2 sheep) = 748/1140=187/285 = 0,6561, so the probablility to choose 2 sheep y 65,61%

Bharat B Sharma

There are 12 sheep and 8 goats in a field. The farmer selects three animals, at random, to bring to market. What is the probability that the farmer chooses at least two sheep?

Solution- Probability is 0.656.

To find out probability based on total combinations possible:

Given:

Sheep – 12, Goats 8; Total animals 20

To select 3

To find probability of having at least two sheep ( 2 sheep  or 3 sheep combinations)

A-- Combination of Exactly two sheep--sheep are chosen:

Number of ways to choose 2 sheep out of 12

                 = (Factorial 12)/(Factorial 2 x Factorial 10) = (12x11)/(2x1) = 132/2 = 66

Number of ways to choose 1 goat out of 8 

                 = (Factorial 8)/(Factorial 1 x Factorial 7)  = 8/1=8

 Therefore – Total combinations possible are 66x8 == 528  ---A

B -- Similarly Combination of selecting exactly three sheep

= (Factorial 12)/(Factorial 3 x Factorial 9) = (12x11x10)/(3x2x1) = 220  --  B

So - Total combination of  minimum two sheep

=A+B  528+220 = 748

Total combinations for selecting 3 animals == (Factorial 20)/(Factorial 3xFactorial 17)

=(20x19x18)/(3x2x1) = 1140

Therefore, total probability of selecting minimum of 2 sheeps is

748 / 1140 = 187/285 =0.656

Kamila Staszewska

2 sheep and 1 goat: 12!/(2!*10!) * 8!/(1!*7!)= (12*11)/2 * 8 = 66 * 8 = 528

3 sheep: 12!/(3!*9!)=12*11*10/6=220

 All combinations possible – any 3 animals: 20!/(3!*9!)=20*19*18/6=1140

P=(528+220)/1140 = 65.6%.

Sudi Sudarshan

My answer to this week's brainteaser: 0.65614

In order to get at least 2 sheep out of 3 chosen randomly, the possible combinations are: SSG, SGS, GSS, SSS

Probability of at least 2 sheep chosen out of 12 sheep and 8 goats = P(2S) + P(3S)

P(2S) = 3*(12/20)(11/19)(8/18)

P(3S) = (12/20)(11/19)(10/18)

P(2+S) = ((3*12*11*8)+(12*11*10))/(20*19*18) = (12*11*34)/(20*19*18) = 0.65614

Andrew Knox

Possible outcomes are:

S     S     S    with chance 12/20 x 11/19 x 10/18 = 0.193

S     S     G    with chance 12/20 x 11/19 x 8/18 = 0.154

S     G     S    with chance 12/20 x 8/19 x 11/18 = 0.154

G     S     S    with chance 8/20 x 12/19 x 11/18 = 0.154

Total chance of at least 2 sheep =  0.193 + 0.154 + 0.154 + 0.154 = 0.655.

What do these numbers have in common?

484, 676, 10201, 12321

Answer: Everything is back on track, with neat answers now to both this week’s teaser and the one carried over from last week (See further below): For Q3, it’s very well done to: Kamila Staszewska, R&D / quality lead, Abcon Industrial Products Ltd, Cootehill, Co. Cavan, Ireland; John Bowen, consultant, Bromsgrove, UK; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; Bharat B Sharma, technical director, Rajsha Chemicals Pvt. Ltd, (TWC Group), Vadodara (Guj) India; Peter D. Talbot, research scientist, Chem-Trend LP, Howell, Michigan, USA; and everyone else who had a go.

SOLUTIONS

Kamila Staszewska

These numbers are palindromes that can be square-rooted to whole numbers:

484 = 22^2

676=26^2

10201 = 101^2

12321=111^2

John Bowen

These are all palindromic square numbers, i.e. each reads the same forwards and backwards and are [with the exception of 676 which is the square of 26] they are also squares of palindromic numbers. 22,101,121]

Amparo Botella

The four numbers given are palindromes (read the same forwards and backwards)

Also the four numbers have a perfect square:

484 = 22x22

676= 26x26

10201=101x101

12321 = 111x111

Bharat B Sharma

These are part of a special series called — Numbers that are both squares as well as Palindromes  (sequence that reads the same  both forwards as well as reverse (backwords).

The series starts with  0, 1, 4, 9, 121, 484, 676, 10201, 12321, 14641, 40804, 44944.... etc.

So the next number in the series would be 14641. 

Common part is - Numbers are Squares as well as Palindromes!!!

Peter Talbot

These four examples are squares of numbers whose factors are all prime

Viz. 484 = (22)2 = (1 X 2 X 11)2

676 = (26)2 = (1 X 2 X 13)2

10201 = (101)2 = (1 X 101)2

12321 = (111)2 = (1 X 3 X 37)2

New teaser on Tuesday (due to yet another UK bank holiday)

What do US presidents George Washington and James Monroe, Britian's George III and duke of Wellington and Louis XV of France have in common?

Clue: Cities...

Answer: With the above clue we finally got 'correct' answers to this teaser - from the same readers that valliantly attempted to make the elusive connection last week. So. it's very well done again to: Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain; John Bowen, consultant, Bromsgrove, UK; Bharat B Sharma, technical director, Rajsha Chemicals Pvt. Ltd, (TWC Group), Vadodara (Guj) India.

SOLUTIONS

John Bowen

I think that each of these has a capital city named after him:

George Washington:  Washington DC, capital of USA

James  Monroe:  Monrovia, capital of Liberia

George III:  George Town capital of  Cayman Isles

Duke of Wellington:  Wellington, capital of New Zealand

Louis XV of France: Port Louis, capital of Mauritius

Bharat B Sharma

Based on the clue added to last weeks teaser, please note that all these five great leaders have a capital of country named after them:

Washington DC: The capital of the United States is named in honor of the first President, George Washington.

Monrovia: The capital of Liberia is named for the fifth US President, James Monroe.

Georgetown: The capital of Guyana is named after the British monarch, King George III.

Wellington: The capital of New Zealand is named for the victor of the Battle of Waterloo, the Duke of Wellington. 

Port Louis: The capital of Mauritius is named for the French monarch, Louis XV.

Amparo Botella

George Washington → Washington, D.C., the capital of the United States.

James Monroe → Monroe, cities and counties in several U.S. states are named after him (e.g., Monroe, Louisiana; Monroe, Michigan).

George III → Georgetown (e.g., Georgetown in Washington, D.C., and others in the U.S. and former British colonies).

Duke of Wellington → Wellington, the capital city of New Zealand.

Louis XV → Louisville, Kentucky, named in his honor for French assistance during the American Revolutionary War.

ORIGINALLY SUGGESTED SOLUTIONS

Amparo Botella

Maybe too obvious, but they all overlapped in the 18th. century. And all of them were related to revolution in different ways.

John Bowen

I think they all had George as one of their names.

Bharat B Sharma

All these great leaders have lived/ruled during a period of significant political and social upheaval in Europe and the Americas. All were directly or indirectly leaders (both in the political and military sense) during all, or any of the American Revolution, the French Revolution, and the Napoleonic Wars.

George Washington was the commander-in-chief of the Continental Army during the American Revolutionary War, leading the Patriot forces to victory against the British.  He viewed the French Revolution with optimism, recognizing its shared ideals with the American Revolution. He also recognized the probability of instability and violence, fearing that the revolution might not achieve a lasting peace and happiness. He ultimately decided to maintain US neutrality, fearing entanglement in European conflicts and potential economic consequences

James Monroe, the fifth President of the United States, actively participated in the American Revolutionary War, serving in the Continental Army.  His involvement with the French Revolution was based on his diplomatic services and his strong ties to Thomas Jefferson and the Democratic-Republican party, which supported the revolution. He, in 1974, was appointed as the U.S. minister to France, where he secured the release of Thomas Paine and facilitated trade negotiations.

King George III's faced the American Revolution thus facing significant challenges in maintaining control over the colonies. He initially reacted with apprehension to the French Revolution and its aftermath. He finally took strong measures against any radical elements within Britain. He ultimately sided with the forces resisting revolutionary France, leading Britain into war with France.

The Duke of Wellington, Arthur Wellesley, played a significant role in the Napoleonic Wars, a direct consequence of the French Revolution. He witnessed the early years of the Revolutionary Wars, later rose to prominence in the Peninsular War and actively participated and led the allies to victory at Waterloo

Louis XV, who reigned before the American Revolution, played a role in the events that led to the conflict and the French Revolution. He was not directly involved in the American Revolution, but his leadership resulted in the weakening of French power in North America and the growing discontent with the French monarchy, which ultimately fuelled  the revolutionary movements in both America and France.

Trainspotters Adam and Brian stand back-to-back next to a railway line. When the front of a train passes them, Adam starts walking in the opposite direction of the train, while Brian walks in the direction of the train. They walk at exactly the same speed, stopping precisely when the back of the train passes. If Adam walks exactly 40 metres, and Brian walks exactly 50 metres, how long is the train?

Answer: Right on track this week were: Bharat B Sharma, technical director, TWC Group (Rajsha Chemicals Pvt. Ltd), Vadodara (Guj) India; Andrew Knox, Rubbond International, Ohé en Laak, The Netherlands; Sudi Sudarshan, principal consultant, Global Mobility Strategies, USA; Amparo Botella, responsable de Compras y Calidad, Ismael Quesada SA, Elche, Alicante, Spain. Well done to our select group, and everyone else who had a go. 

SOLUTIONS

Bharat B Sharma

When the back of the train passes Adam, he and Brian have both walked 40meters.

    Meaning Brian will walk 10 meters more when the train passes him. (50 m-40 m).

    40 meters walked by Adam

    50 meters walked by Brian

  Total 90 meters that the back of the train travels in the same time that Brian walks 10 meters.

This means train is 90/10 (or 9 times) faster v/s. Adam and Brian.

In 40 meters walked by Adam, train will move 40x9 = 360meters (front of the train). Plus Alans distance of 40 meter (to reach end of the train) making total length of the train = 360+40 = 400 Meters.

Andrew Knox

If the train is travelling at speed V m/s and Adam and Brian both walk at the same speed v m/s then vt1 = 40 m or t1 = 40/v for Adam and t2 = 50/v for Brian. If the train is L m long, then equations are V = (L-40)/40/v = (L+50)/50/v. Solving for L gives L/40 - L/50 = 2, so 1.25 L - L = 100, or L = 400 m.

Sudi Sudarshan

Let s be the speed of Adam and Brian, L the train length and t the time for the back of thr  train to pass Adam

When the back of the train passes Adam, he and Brian have already walked 40m and so the distance between them is 80m. In order for the back of the train to catch up with Brian it needs to travel 90m in the same time that Brian walks 10m.

So, trains speed = 9s

Time taken by Adam until the back of the train passed him = t = distance/time = 40/s

The train traverses (L-40 ) m during the same time at a speed of 9s

(L-40)/9s = 40/s

Cancelling s from both sides and simplifying,

L-40 = 360

L = 360+40 = 400 metres

Amparo Botella

Train speed = v

Walking speed = w

Time taken until the end of the train passes = t

For Adam: relative speed = v + w

t = v + w

And he walks 40 metres in that time:

40 = w  t = v + w

For Brian (same direction): relative speed = v - w

t = v - w

And he walks 50 metres in that time:

50 = w t = v - w

40(v + w)=50(v - w)

40v + 40w = 50v - 50w

40w + 50w = 50v - 40v

90w = 10v v=90w/10 = 9W

L = 40(9w + w) = 400.

 

New teaser on Monday