*Binomial Distributions

Question 1

What is a Bernoulli Distribution?

Answer 1

Any probability distribution with only 2 possible results, the desire result (x=1) and the undesired result (x=0)

Question 2

In what 3 ways Bernoulli Distribution be represented?

Answer 2

1. Piecewise function, the thing with the { and the two functions
2. Probability Distribution Table, a table with two rows and three columns top row is x-values, bottom row is probability values
3. Probability Bar Graph, x-values on the bottom, probability values on the side

Question 3

What is p?

Answer 3

The probability of the desired result

Question 4

What is q?

Answer 4

The probability of the undesired result

OR

q = 1 - p

Question 5

Given that nothing extra is said, what is the mean?

Answer 5

E(x) = p

Question 6

Given that nothing extra is said, what is the Variance?

Answer 6

V(x) = pq

OR

V(x) = p(1 - p)

Question 7

What is the standard deviation and how is it found?

Answer 7

√V(x) = √(pq) = σ

The spread of the data

Question 8

Given that we are told the number of trials (n) and the number of wanted results (r), the probability can be found by-

Answer 8

Binomial Formula: nCr • p^r • q^(n - r)

Question 9

Drawing a probability histogram

1. First step?

Answer 9

Locate the n and the p
Make r and q

E.g. Draw a histogram for the probabilities of rolling a 2 on a dice 5 times
n = 5, p = 1/6

If this is the case,
• r = 0, 1, 2… n
• q = 5/6

Question 10

Drawing a probability histogram

2. Second step?

Answer 10

Find probabilities with the Binomial Formula for all values (0 → n = r-values)

E.g. Draw a histogram for the probabilities of rolling a 2 on a dice 5 times
Do nCr p^r q^(n-r) for r = 0, 1, 2, 3, 4, 5

Question 11

Drawing a probability histogram

3. Third step?

Answer 11

Put r-values on the x-axis, put probability on the y-axis

E.g. Draw a histogram for the probabilities of rolling a 2 on a dice 5 times
Draw the graph

Question 12

Drawing a probability histogram

4. Fourth step?

Answer 12

Label its shape, check if it’s a slide

If there’s isn’t a slide and it’s even → symmetrical
If there is a slide
- If it’s fun (short to long) → positive skew
- If it’s not fun (long to short) → negative skew

E.g. Draw a histogram for the probabilities of rolling a 2 on a dice 5 times
Here, there’s no slide and it’s even, thus symmetrical

Question 13

Given that we’re given a number of trials, what is the mean?

Answer 13

E(x) = np

Question 14

Given that we’re given a number of trials, what is the variance?

Answer 14

V(x) = npq

OR

V(x) = np(1 - p)

note with the same standard deviation formula √V(x)

Question 15

A sample population is?

Answer 15

A small portion of the total population, given as s

Question 16

An observation (desired result) is?

Answer 16

The number of those with a desired trait in the small population, given as d

Question 17

A sample proportion is?

Answer 17

The probability of finding a desired trait within a small portion, given as p^ = d/s

Question 18

Given that we’re told it’s a sample, what is the mean?

Answer 18

E(x) = p = p^ = d/s

Question 19

Given that we’re told it’s a sample, what is the variance?

Answer 19

V(x) = p^q^/s

Question 20

Finding the probability of a sample proportion

1. First step?

Answer 20

Find the z score, approximate to 2 decimals

Question 21

Finding the probability of a sample proportion

2. Second step?

Answer 21

Check the table for the t-score:
If it’s less than question → straight to the t-score
If it’s greater than question → 1 minus the t-score
If it’s between question → bigger t-score minus the smaller t-score

• note* table is in page 633-34
• note* only works for normal distributions

Question 22

A z-score can be found with the formula, and WHATS so special about this one (binomials)?

Answer 22

z = (x - μ)/σ

x = probability of a given event happening
μ = p^ = d/s
σ = √(p^q^/s)

Question 23

If we are given p^, what is q^?

Answer 23

q^ = 1 - p^