The probability of the defective fitness equipment being taken last

[nobox]

Suppose there are 10 pieces of fitness equipment in a gym. One of them is slightly defective but usable and all the other 9 are fine. Suppose the gym has a large number of patrons, say 1,000 and only 10% of them are aware of the defective one therefore would avoid picking it unless it is the last available one. All other patrons would pick equipment randomly. Suppose there is no exchange after equipment is picked, and patrons arrive randomly, what is the likelihood that the defective equipment becomes the last one to be picked? It has economic ramification in the market place. We can easily expand it to stock market or a supermarket.

[imatfaal]

The chance that it will be last chosen is the chance that all nine of the first nine will not choose it.

 

The Chance the first one will chose it is 900/1000 * 1/10 ie chance it is a clueless patron times the probability of picking the dodgy machine.

 

The chance that the first one will not chose is 1 - 900/1000 * 1/10

 

The chance that the second one will not chose is 1 - 899/999 * 1/9

ie Total prob minus chance user is still clueless and picks dodgy machine from 9 remaining

 

etc

 

You get nine sums like that - multiply all together and that is the probability that none of first nine choose the machine which is the same as the probability that the tenth will

 

it is about 13.2 pct