Poisson Distrib: Prob 10 Tubes Show Growth

[Vagrant]

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*m^r)/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?

 

[Mark44]

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*m^r)/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.

 

[Vagrant]

Taking p=0.9502 as a success, n=10 for binomial distribution:
P=¹⁰C₁₀*p¹⁰*q⁰
P=0.600.

Is this correct?

 

[Dick]

Taking p=0.9502 as a success, n=10 for binomial distribution:
P=¹⁰C₁₀*p¹⁰*q⁰
P=0.600.

Is this correct?

Yes.

 

[Vagrant]

Thanks.

 

[x12179x]

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)?

 

[Mark44]

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.