SOLUTION: A urn contains 10 coins of which 4 are counterfeit. Coins are removed from the urn, one at a time, until all counterfeit coins are found. If the random variable X denotes the num

Algebra ->  Probability-and-statistics -> SOLUTION: A urn contains 10 coins of which 4 are counterfeit. Coins are removed from the urn, one at a time, until all counterfeit coins are found. If the random variable X denotes the num      Log On


   

Question 1149232: A urn contains 10 coins of which 4 are counterfeit. Coins are removed
from the urn, one at a time, until all counterfeit coins are found. If the
random variable X denotes the number of coins removed to find the first
counterfeit one, then what are the space and the probability density function
of X?
Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


X=1: You need to get a counterfeit on the first draw.
P%281%29+=+4%2F10

X=2: You need to get a genuine first and a counterfeit second.
P%282%29+=+%286%2F10%29%284%2F9%29

X=3: You need to get a genuine on the first two draws and a counterfeit on the third.
P%283%29+=+%286%2F10%29%285%2F9%29%284%2F8%29

Similarly....

P%284%29+=+%286%2F10%29%285%2F9%29%284%2F8%29%284%2F7%29
P%285%29+=+%286%2F10%29%285%2F9%29%284%2F8%29%283%2F7%29%284%2F6%29
P%286%29+=+%286%2F10%29%285%2F9%29%284%2F8%29%283%2F7%29%282%2F6%29%284%2F5%29
P%287%29+=+%286%2F10%29%285%2F9%29%284%2F8%29%283%2F7%29%282%2F6%29%281%2F5%29%284%2F4%29

Simplify as required to find the probability density function.

It is interesting to observe that the sum of all the probabilities is indeed 1.

The sample space is 1 to 7. Clearly if there are 6 genuine coins and 4 counterfeit coins, the "worst case" if you are trying to get a counterfeit coin is to draw all 6 of the genuine coins first, thus drawing the first counterfeit coin on the 7th draw.