Probability

Question 1

for overlapping venn diagrams, what does P(A u B) equal

Answer 1

P(A u B) = P(A) + P(B) - P(A n B)

Question 2

what does it mean if events A and B are mutually exclusive

Answer 2

A and B cant happen at the same time, AKA there’s no P(A n B)

Question 3

what does P(A u B) equal for mutually exclusive events

Answer 3

P(A u B) = P(A) + P(B)

Question 4

for independent events, what does P(A n B) equal

Answer 4

P(A n B) = P(A) + P(B)

Question 5

how do you denote the probability of A happening given that B happens

Answer 5

P(A|B)

Question 6

what does P(A|B) equal

Answer 6

P(A|B) = P(A n B) / P(B)

Question 7

what is the relationship between the events A and B if P(A|B) = P(A|B’) and why (B’ = not B)

Answer 7

• the events A and B are independent

- because the probability of A happening if not influenced by whether B happens

Question 8

what is the permutations formula for permuting r objects from a total of 10

Answer 8

nPr = n! / (n - r)!

Question 9

when do you use the permutations formula

Answer 9

when the order that events happen matters

Question 10

what is the combinations formula for combining r articles from n

Answer 10

nCr = n! / [(n - r)!*r!]

Question 11

what is the formula for the mean, μ, of a total sample size N

Answer 11

μ = Σ(i = 1 to N) of x(i) / N

Question 12

using the same symbols for the formula of the mean, what is the formula for the variance, σ^2

Answer 12

σ^2 = Σ(i = 1 to N) of (x(i) - μ)^2) / N

Question 13

what is the formula for the standard deviation, σ

Answer 13

the square root of the variance

Question 14

how would you calculate what percentage of data lies within x standard deviations of the mean, x being an arbitrary number

Answer 14

• 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

Question 15

what is the formula for the mean, m, of an actual sample (fraction of the total sample) with size n

Answer 15

m = Σ(i = 1 to n) of x(i) / n

Question 16

using the same symbols for the formula of the mean, what is the formula for the variance of an actual sample, s^2

Answer 16

s^2 = Σ(i = 1 to n) of (x(i) - m)^2) / (n-1)`

Question 17

what is the formula for the standard deviation of an actual sample, s

Answer 17

the square root of the variance

Question 18

what is the formula for estimating the sample standard deviation without using the mean

Answer 18

s = sqrt{[(Σ(i = 1 to n) of x(i)^2) - 1/n*(Σ(i = 1 to n) of x(i))^2)] / n-1}

Question 19

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

Answer 19

μ = Σ(j = 1 to M) of x(j)*P(x(j))

Question 20

using the same symbols for the formula of the mean in the discrete probability distribution, what is the formula for the variance, σ^2

Answer 20

σ^2 = Σ(j = 1 to M) of [(x(j) - μ)^2]*P(x(j))

Question 21

what is the formula for the standard deviation, σ, in the discrete probability distribution

Answer 21

the square root of the variance

Question 22

for a continuous probability distribution f(x), what is the formula for the probability of (a < x < b)

Answer 22

P(a < x < b) = int[f(x)] dx from limits b to a

Question 23

what is the formula for the mean, μ, of a continuous probability distribution

Answer 23

μ = int[x*f(x)] dx from +∞ to -∞

Question 24

what is the formula for the variance, σ^2, of a continuous probability distribution

Answer 24

σ^2 = int[((x - μ)^2)*f(x)] dx from limits +∞ to -∞

Question 25

what is the formula for the standard deviation, σ, of a continuous probability distribution

Answer 25

the square root of the variance

Question 26

what does +∞ to -∞ really mean in the practical sense

Answer 26

the limits encompass all the possible values with a probability of happening

Question 27

how large does a sample size n generally need to be in order for the mean x̄ to be assumed to be normally distributed

Answer 27

n > 30

Question 28

what is the relationship between the standard deviation of x̄, s(x̄), and the standard deviation of the original experimental data

Answer 28

• 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

Question 29

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

Answer 29

0.67s(x̄)

Question 30

what is the scalar multiple of s(x̄) for x̄ to lie within the true value 68% of the time

Answer 30

1s(x̄)

Question 31

what is the scalar multiple of s(x̄) for x̄ to lie within the true value 95% of the time

Answer 31

2s(x̄)

Question 32

what is the scalar multiple of s(x̄) for x̄ to lie within the true value 99.73% of the time

Answer 32

2s(x̄)