A Problem of Probability

        This happened when I was visiting Bagaio, a holiday resort in the Philippines.

        I was lunching with a young mathematics professor who was also holidaying. We were seated at a table by the window. We spoke about various things and finally we hit upon the subject of the determination of the probability of a coincidence.

       The professor, incidentally his name is Prof. Alfredo Garcia – he took out a coin and said ‘Now watch, I am going to flip this coin and said ‘Now watch, I am going to flip this coin on the table without looking. Tell me what’s the probability of a tail – up turn?

At that time two other friends of ours walked in and joined us at the table for coffee. We briefly explained to them the topic of our discussion.

‘First of all, professor, explain to us what is ‘probability’ said one of the two friends’.

‘Yes, please do. Not everyone knows it’ I said.

‘Certainly. Well it’s really very simple. If I toss a coin in the air, there are only two possible ways in which it would fall. Head or tail. Of these only one will be exactly what we want. Let’s call it a favorable occurrence. Then it can be deduced mathematically this way.

The number of favorable occurrence =1

The number of possible occurrence =2

In this way the probability of a tail-up can be represented by the fraction ½.

What if it is not a coin …..? Say it is something more complicated ‘interrupted one of the friends.

‘Say for example a die ‘joined in the other.

‘Yeah, a die. Let me see …. Of course …. It’s cubical in shape’.

 The professor was thoughtful for a moment. The n he continued ‘Yeah … it has numbers on each of its faces …. 1, 2, 3,4,5,6…

‘Now what the probability …… is of …….say number 6 turning up? I asked.

‘Well, there are six faces. Therefore we have to see how many possible. Occurrences there are said the professor.’ Any of the numbers from 1 to 6 can turn up.

The favorable occurrence for us naturally will be 6.

And naturally the probability in this case will be 1/6.

‘But can you really compute in this manner any event? Queried one of the friends. ‘Take for example if I were to bet that the very next person to pass our window will be a woman, what’s the probability that I would win the bet?

‘Well, I would say it is ½, provided we decide to regard even a little boy as a man and a little girl as a woman, ‘the professor replied.

‘That’s assuming that there are an equal number of men and women in the world’ I joined in.

‘In such a case what’s the probability that the first wo persons passing the window will be men?’ one of the friends asked.

‘Well, a computation of this kind will be a little more complicated. We’ll have to try all the possible combinations. The first possibility will be that both the persons will be men. Second possibility that the first person may be a man and second a woman. Third, the first person may be a woman and second a man. Fourth, both the persons may be women. That makes four combinations. And of course, of these four combinations only one is favorable’.

‘I see ‘I agreed.

‘So, the probability is ¼ the professor continued. 

‘And that’s the solution to your problem’.

We were all silent for a moment. Then one of the friends spoke.

‘Supposing instead of two we think of three men.

What would be the probability that the first three persons to pass our window will be men?

‘The solution is obvious, isn’t it? The professor said. ‘We start by computing the number of possible the combinations. When we calculated for two passersby, the number of combinations we found was 4. Now by adding one more passers-by we have to double the number of possible combinations, because each of the four groups of the two passers-by can be joined by a man or a woman. That makes the number of possible combinations in this case 4x2 =8.’

‘One would have never thought of it that way’ remarked a friend.

‘But you see it for a fact! The probability is quiet obvious – it is 1/8. Only one in eight will be a favorable occurrence. The method of calculating the probability is very easy really.’

The professor took out a ball pen from his pocket and wrote on the white table cloth.

‘Two passers by the probability is ½ x ½ = ¼ OK’

‘Yes’ I agreed.

‘For three it is ½ x ½ x½ =⅛ ‘

‘Agreed’ said a friend.

‘Now for four the probability will be, naturally, the product of 4 halves, that is 1/16.’

‘It appears the probability grows less each time’ remarked a friend.

‘Right. Take the case of ten passersby, for example.

The answers will be the product of 10 halves. Do you know how much it is 1/1024?’

‘No’ we all said in a chorus.

‘In other words said one of the friends ‘if I bet a dollar that the first ten passers- by will be all men, chances of winning the bet is only 1/1024.’

‘Well I can put up a bet for a $ 100.00 that it will not happen’. The professor said very confidently.

‘I can surely use a hundred dollars. Wish I could catch you on that bet professor’ I joined.

‘But then your chance to win will be only one in one thousand twenty four’

‘I don’t mind………..all I would be losing is only $ 1.00’

‘Still a dollar is dollar’ said a friend.

‘And your chances of winning the bet is so remote’ said the other.

I looked out of the window. The road was somewhat deserted. After lunch, most people were home deserted. After lunch, most people were home enjoying their after – lunch siestas. I looked at my watch. It was almost approaching 2 p.m. – only a minute or left. I spoke quickly.

‘Tell me professor, what are the chances of my winning. If I were to bet one dollar against one hundred that the next hundred passers by outside our window will be men’

‘ your chances of winning would be even less than one in a million for twenty passers- by and for 100 passers- by the probability would be even less than’ he wrote on the table cloth

1 /1000 000 000 000 000 000 000 000 000 000

‘Really!’ I remarked ‘I still want to take the bet with you that the next one hundred passers- by will be men’.

‘I will give you one thousand dollars instead of one hundred. You can’t win the bet’ he said excitedly.

I looked at my watch again. It was only a few seconds before 2 P.M.

As our friends watched us with utter amusement I reached for the professor’s hand and shook it, confirming the bet. The very next moment something happened.

In exactly five minutes after that the professor was heading towards the bank to encase all his travelers’ cheques in order to pay me my one thousand dollars.

How do you think I won my bet?

Answer:    A battalion of soldiers came marching past our window. I had known all along that a corps of army cadets consisting of over 100 men marched past our hotel exactly at 2 P.M. every Tuesday afternoon.