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Poisson Distribution

  • Thread starter Vagrant
  • Start date
  • #1
195
1

Homework Statement



A source of liquid is known to contain bacteria, with the mean number of bacteria per cubic centimeter equal to 3. Ten 1 c.c. test tubes are filled with liquid. Calculate the probability that all 10 test tubes will show growth, that is contain at least 1 bacterium each. (use Poisson distribution)

Homework Equations


P(r)=(e-m*mr)/r!

The Attempt at a Solution


Taking m=3;
N=10 (not used)
P(1)+P(2)+P(3)=0.5974

Answer provided is 0.600, is my solution correct?
 

Answers and Replies

  • #2
33,181
4,860

Homework Statement



A source of liquid is known to contain bacteria, with the mean number of bacteria per cubic centimeter equal to 3. Ten 1 c.c. test tubes are filled with liquid. Calculate the probability that all 10 test tubes will show growth, that is contain at least 1 bacterium each. (use Poisson distribution)

Homework Equations


P(r)=(e-m*mr)/r!

The Attempt at a Solution


Taking m=3;
N=10 (not used)
P(1)+P(2)+P(3)=0.5974

Answer provided is 0.600, is my solution correct?
No. The probability of at least 1 bacterium in a test tube is P(1) + P(2) + P(3) + P(4) + P(5) + ... Why did you stop at 3 bacteria?

The probability I showed is equal to 1 - P(0), which is about .9502. This is the probability of finding 1 or more bacteria in one test tube. Now, how do you get the probability of finding 1 or more bacteria in all 10 test tubes?

The answer I got was .6001, rounded to 4 decimal places.
 
  • #3
195
1
Taking p=0.9502 as a success, n=10 for binomial distribution:
P=10C10*p10*q0
P=0.600.

Is this correct?
 
  • #4
Dick
Science Advisor
Homework Helper
26,258
618
Taking p=0.9502 as a success, n=10 for binomial distribution:
P=10C10*p10*q0
P=0.600.

Is this correct?
Yes.
 
  • #5
195
1
Thanks.
 
  • #6
27
0
P=10C10*p10*q0
I'm working on a similar problem. Can someone tell me where this equation came from? What's c and q (and what do those 2 10's next to the c mean)?
 
  • #7
33,181
4,860
C stands for combinations. The first expression in this binomial probability is 10C10, which is the number of combinations of 10 things taken 10 at a time, which turns out to be 1. p is a probability of something happening (you didn't give any context here, so I can't say anything more) and q is the probability of something not happening, which means that q = 1 - p.
 

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