Lecture 6: Probability Distributions, Binomial Distribution

Physics 3719Spring Semester 2011

The probability distribution for an excited state of a particle confined to a region with hexagonal symmetry

C Gray et al 2004

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Experimental vs Parent Distributions

● Experimental: If I make n measurements of a quantity x, they can be sorted into a histogram to determine the experimental distribution.

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Experimental vs Parent Distributions

● Experimental: If I make n measurements of a quantity x, they can be sorted into a histogram to determine the experimental distribution.

● If I divide the number of events in each bin by the total number of events, I have an experimental probability distribution.

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Experimental vs Parent Distributions

● If I make n measurements of a quantity x, they can be sorted into a histogram to determine the experimental distribution.

● If I divide the number of events in each bin by the total number of events, I have an experimental probability distribution.

● The parent probability distribution is the distribution we would see as n

tot → infinity.

● The physics lies in the parent distribution, which we must try to infer...

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The 3 (most?) Important Probability Distributions

● Binomial: Result of experiment can be described as the yes/no or success/failure outcome of a trial. The probability of obtaining success is known.

● Poisson: Predicts outcome of “counting experiments” where the expected number of counts is small.

● Gaussian: Predicts outcome of “counting experiments” where the expected number of counts is large.

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The 3 (most?) Important Probability Distributions

● Binomial: Result of experiment can be described as the yes/no or success/failure outcome of a trial. The probability of obtaining success is known.

● Poisson: Predicts outcome of “counting experiments” where the expected number of counts is small. (Special case of binomial.)

● Gaussian: Predicts outcome of “counting experiments” where the expected number of counts is large. (Special case of binomial.)

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Binomial Distribution

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Example: If I toss a coin 3 times, what is the probability of obtaining 2 heads?

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Example: A gambler rolls 4 6-sided dice. What is the probability...

a) that exactly two have 1 facing up?b) that all have 1 or 2 facing up?c) that one or more have 3,4,5, or 6 facing up?

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Example: If I toss n = 10 coins 100 times (a) what is the mean number of heads? (b) what is the standard deviation of the number of heads observed?

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Example: A hospital admits four patients suffering from a disease for which the mortality rate is 80%. Find the probabilities that (a) none of the patients survives (b) exactly one survives (c) two or more survive.

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Example: In a scattering experiment, I count forward- and backward scattering events. I expect 50% forward and 50% backward.

What I observe:

TK

472 back scatter 528 forward scatter

What uncertainty should I quote?