## What’s the chance that you ever take the lead?

Posted in Cute Problems on January 25, 2014 by telescoper

Here’s one of an occasional series of cute problems, which I offer as a challenge for whiling away a wild and rainy Saturday afternoon..

You enter a competition which consists of a never-ending series of contests. The probability that you win any single contest is p, and the outcomes of the contests are independent of one another.

Let X be the probability that you ever take the lead in the competition. What is X in terms of p, for any value of p?

UPDATE: Since a correct answer has now been posted, here is my solution:

Consider the first contest: the probability that you win it is p and if you do you take the lead straight off.

If you lose the first one, with probability (1-p), you are down by 1. Now you must (a) make up the deficit and (b) go on to take the lead. Clearly the probability of (a) is just X (the same as getting ahead from a level start). The probability of (b) is also X.
Hence X=p+(1-p)X^2.

There are two solutions of this quadratic equation: X=1 and X=p/(1-p). But the answer must be a probability so cannot exceed unity. Hence if p>1/2 then X=1, in accord with intuition: in the long run you’d expect to lead sometime if p>1/2. If p<1/2 then the other solution is correct. The two solutions match at p=1/2.

Simples.

## The January Man

Posted in Uncategorized with tags , on January 25, 2014 by telescoper

The January man he walks the road
In woollen coat and boots of leather
The February man still shakes the snow
From off his hair and blows his hands
The man of March he sees the Spring and
Wonders what the year will bring
And hopes for better weather

Through April rains the man comes down
To watch the birds come in to share the summer
The man of May stands very still
Watching the children dance away the day
In June the man inside the man is young
And wants to lend a hand
And grins at each new colour

And in July the man in cotton shirt
He sits and thinks on being idle
The August man in thousands take the road
To watch the sea and find the sun
September man is standing near
And Autumn is his bridle

The man of new October takes the reins
And early frost is on his shoulder
The poor November man sees fire and rain
And snow and mist and wintery gale
December man looks through the snow
To let eleven brothers know
They’re all a little older

And the January man comes round again
In woollen coat and boots of leather
To take another turn and walk along
the icy road he knows so well
For the January man is here for
Starting each and every year