Probability Flashcards
for overlapping venn diagrams, what does P(A u B) equal
P(A u B) = P(A) + P(B) - P(A n B)
what does it mean if events A and B are mutually exclusive
A and B cant happen at the same time, AKA there’s no P(A n B)
what does P(A u B) equal for mutually exclusive events
P(A u B) = P(A) + P(B)
for independent events, what does P(A n B) equal
P(A n B) = P(A) + P(B)
how do you denote the probability of A happening given that B happens
P(A|B)
what does P(A|B) equal
P(A|B) = P(A n B) / P(B)
what is the relationship between the events A and B if P(A|B) = P(A|B’) and why (B’ = not B)
- the events A and B are independent
- because the probability of A happening if not influenced by whether B happens
what is the permutations formula for permuting r objects from a total of 10
nPr = n! / (n - r)!
when do you use the permutations formula
when the order that events happen matters
what is the combinations formula for combining r articles from n
nCr = n! / [(n - r)!*r!]
what is the formula for the mean, μ, of a total sample size N
μ = Σ(i = 1 to N) of x(i) / N
using the same symbols for the formula of the mean, what is the formula for the variance, σ^2
σ^2 = Σ(i = 1 to N) of (x(i) - μ)^2) / N
what is the formula for the standard deviation, σ
the square root of the variance
how would you calculate what percentage of data lies within x standard deviations of the mean, x being an arbitrary number
- multiply the standard deviation σ by x
- add and subtract this number to / from the mean to get the mean range
- count the number of data values that lie within this range
- calculate the percentage relative to all the data values
what is the formula for the mean, m, of an actual sample (fraction of the total sample) with size n
m = Σ(i = 1 to n) of x(i) / n
using the same symbols for the formula of the mean, what is the formula for the variance of an actual sample, s^2
s^2 = Σ(i = 1 to n) of (x(i) - m)^2) / (n-1)`
what is the formula for the standard deviation of an actual sample, s
the square root of the variance
what is the formula for estimating the sample standard deviation without using the mean
s = sqrt{[(Σ(i = 1 to n) of x(i)^2) - 1/n*(Σ(i = 1 to n) of x(i))^2)] / n-1}
for a discrete probability distribution P(x) where each value x(j) has a probability P(x(j)), what is the formula for the arithmetic mean, μ, where x(j) represents each different value possible and M represents the number of different possible values
μ = Σ(j = 1 to M) of x(j)*P(x(j))
using the same symbols for the formula of the mean in the discrete probability distribution, what is the formula for the variance, σ^2
σ^2 = Σ(j = 1 to M) of [(x(j) - μ)^2]*P(x(j))
what is the formula for the standard deviation, σ, in the discrete probability distribution
the square root of the variance
for a continuous probability distribution f(x), what is the formula for the probability of (a < x < b)
P(a < x < b) = int[f(x)] dx from limits b to a
what is the formula for the mean, μ, of a continuous probability distribution
μ = int[x*f(x)] dx from +∞ to -∞
what is the formula for the variance, σ^2, of a continuous probability distribution
σ^2 = int[((x - μ)^2)*f(x)] dx from limits +∞ to -∞
what is the formula for the standard deviation, σ, of a continuous probability distribution
the square root of the variance
what does +∞ to -∞ really mean in the practical sense
the limits encompass all the possible values with a probability of happening
how large does a sample size n generally need to be in order for the mean x̄ to be assumed to be normally distributed
n > 30
what is the relationship between the standard deviation of x̄, s(x̄), and the standard deviation of the original experimental data
- the s.d. of x̄ is a factor of the sqrt(n) less than the s.d. of the original data
- so in the s.d. formula for a sample with n-1 denominator, you would multiple that by n for the s.d. of x̄
for a certain percentage of the time, x̄ will lie within a + or - scalar multiple of s(x̄) of the true value. what is this scalar multiple for x̄ to lie within the true value 50% of the time
0.67s(x̄)
what is the scalar multiple of s(x̄) for x̄ to lie within the true value 68% of the time
1s(x̄)
what is the scalar multiple of s(x̄) for x̄ to lie within the true value 95% of the time
2s(x̄)
what is the scalar multiple of s(x̄) for x̄ to lie within the true value 99.73% of the time
2s(x̄)
