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| Given n distinct objects, the number of distinct ways you can order them is what? |
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| General formula for n choose k |
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| What is anything choose 0? |
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| Difference between combination and permutation |
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With a combination (subset), we don't care about the order.
With a permutation (ordered subset), we do. |
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| The number of subsets having exactly k elements from a set of n is |
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| Number of distinct subsets a set with n elements has |
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| The trick for binomial theorem (same as in Genetics) |
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(x+y)^n
First there are going to be n+1 terms
The first and last term will have 1 as the coefficients
To figure out the exponents, make the x's start at n and go down. Make the y's start at 0 and go up
To figure out the coefficient for the next term, it's the (coefficient*exponent of x)/term number |
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| a set that represents all possible outcomes of an event, denoted by omega |
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| If Ω is an outcome space and B is some event, then P(B) is |
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| What is a distribution P on omega (the outcome space)? |
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A distribution is any function on the subsets of the outcome space that follows these 3 rules:
For any subset B of the outcome space:
1. P(B) is nonnegative
2. P(omega) = 1. This is true because you're dividing the outcome space by the outcome space if you think about the formal definition of P(B)
3. P(B) equals the union of disjoint subsets of B: P(B1)+P(B2)+P(B3) etc |
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| How many elements does the outcome space for two dice have? |
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| 36. One die has 6 sides so if you have two of those you would square it |
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| An outcome space with a distribution on it |
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| Let A be a subset of Ω. A^c means all the elements in Ω that aren’t in A. So P(A^c) = 1 - P(A) because P(omega) equals P(A) + P(A^c) |
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| P(A U B) = P(A) + P(B) – P(AB) |
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| If A is a subset of B then: P(BAc) = P(B) – P(A) |
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Let Ω be the set consisting of {0, 1}.
Then the Bernoulli distribution is any distribution on Ω.
The probability of subset {1} is p and the probability of the subset {0} is 1-p |
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| If S is a subset of numbers, then P(S) = #observations in the sample S/n |
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means you’re trying to find the probability of A given B
P(A|B) = P(AB)/P(B) |
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What is the probability that both A and B happen?
P(AB) = P(B)P(A|B)
P(AB) = P(A)P(B|A) |
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If P(AB) = P(A)P(B) then A and B are independent
Independence is the concept that one event has no bearing on the probability of another event.
P(A|B) = P(A) and P(B|A) = P(B) |
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Lets us write:
P(A) = P(A|B)P(B) + P(A|B^c)P(B^c) |
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We want p to represent the probability of success, and q to represent the probability of failure
The probability of any particular result, say: {success, failure, failure, success} will be the product of the probability of the first outcome times the probability of the second outcome, etc |
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| The binomial distribution with probability p of success: |
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Applicable for Bernoulli trials:
P(k successes) = (n k)p^kq^(n-k) |
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| Expected number of successes |
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n*p, the number of trials times the probability of a success
Also called the mean and u |
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| Standard deviation of the probability distribution for the number of successes: |
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| Probability of one point is always |
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| Expected value of a normal curve |
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| Standard deviation for standard normal probability |
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| Cumulative Distribution Function |
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Use this if interval is already given
P([a, b]) = O(b) - O(a) |
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| Approximation of Bernoulli histogram by a normal distribution |
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Use this to find intervals which will allow you use the CDF function:
O([b + 1/2 - mean]/standard deviation - O([a - 1/2 - mean]/standard deviation) |
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The number k of success will fall in the range: [expected value - 4*standard deviation, expected value + 4*standard deviation]
This simplifies to [p- 2/square root of n, p + 2/square root of n |
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