1. Physics 270

2. o Experiment with one die o m – frequency of a given result (say rolling a 4) o m/n – relative frequency of this result o All throws are equally probable. o The throws are independent. The measured relative frequency tends to a limit – 1/6.

5.  deals with the probability of several successive decisions, each of which has two possible outcomes  If an event has a probability, p, of happening, then the probability of it happening twice is p 2, and in general p n for n successive trials. If we want to know the probability of rolling a die three times and getting two fours and one other number (in that specific order) it becomes:

6.  However this is only sufficient for problems where the order is specific. If order is not important in the above example, then there are 3 ways that 2 rolls of four and 1 other could occur:  110, 101, 011, where 1 represents a roll of four and 0 represents a non-four roll.  Since there are 3 ways of achieving the same goal, the probability is 3 times that of before, or 6.9%.

9. General Formula  completely specified by n and a  expectation value:  standard deviation:  = (nab) 1/2

10. Example: A baseball player’s batting average is 0.333. The probability for a hit: a = 0.333 The probability for an out: b = 1  a = 0.667 Average number of hits the player gets in 100 at bats µ = na = (100)(0.333) = 33 The standard deviation for 100 at bats So, we can expect, 33 ± 4.7 hits for every 100 at bats

11. If the probability p is small and the number of observations is large the binomial probabilities are hard to calculate. In this instance it is much easier to approximate the binomial probabilities by poisson probabilities. The binomial distribution approaches the poisson distribution for large n and small p.

12. The Poisson distribution can be derived from an approximation of the Binomial distribution.

13. Example: Radioactive Decay n = 10 20 atoms, half-life = 10 12 y = 5 × 10 19 s The probability for a decay: 1/ = 2 × 10  20 /s Average number of decays per second: µ = na = (10 20 atoms)(2 × 10  20 /s) = 2/s What’s the probability of zero decays in one second? What’s the probability of more than 1 in one second?

14. o when there are a large number of events per observation o requires many observations o resembles a normal distribution (It is continuous!) o  is the standard deviation o  is the mean observed number of events

15. The probability of r being in the range r 1 and r 2 is given by which cannot be evaluated analytically, (it can be looked up in a table). If the limits are at +/  ∞, then it normalizes to 1.

16. It is very unlikely (< 0.3%) that a measurement taken at random from a Gaussian data set will be more than ± 3σ from the true mean of the distribution.

17. ● An example illustrating the small difference between the two distributions under the above conditions: Consider tossing a coin 10 000 times. ■ a(heads) = 0.5 ■ n = 10 000 ❒ mean number of heads = μ = na = 5000 ❒ standard deviation σ = [na(1  a)] 0.5 = 50