PSYC 210- Probability Theory and Distributions

If every time an event occurs, it can happen exactly one way at a time_____

elementary event ex. rolling a single dice gives only a single #

event exclusivity

event A and event B = 0 flipping coin can be either heads or tails but not both

collective exclusivity

event A or event B =1 flipping a coin will land either on heads or tails

What is probability distribution?

full set of probabilities

What happens to the normal distribution when standard deviation changes?

higher deviation - more spread/flatter smaller deviation- higher peak

What does a z-score tell us?

how many standard deviations a score is from the mean -allows us to compare score across different distributions

When the outcome of one event does not affect the probability of a second event, they are labeled as ________

independent events ex. throwing two dice

What happens to the binomial distribution when we increase/decrease number of trials?

less trials- more spread more trials- less spread (clumped together b/c the law of independent event)

The normal distribution is defined by 2 measures ...

mean and standard deviation

Let's say you have continuous data, which probability distribution can be used?

normal (most common), T, or F distribution

What's the difference between probability and statistics?

probability- what are the chances/likelihood ... statistics- If someone flips a coin and gets 10 heads consecutively, is it a trick? -we have the data but want to infer the state of the world

What happens to the binomial distribution when the probability is changed?

skews the data

What happens to the normal distribution when mean changes?

the graph shifts either left or right

What is sample space?

the set of all possible outcomes of an experiment ex. flipping a coin S= {HH, TT, HT, TH}

When I flip a coin 5 times, I get 4 heads and 1 tails which is not 50/50. When I flip a coin 10000 times, it is 50/50. Why?

this is due to the law of independent events. the more events that occur, the observed ratios of events will towards the expected ratios of events

Probability of A or B

union P(A) + P(B) - P(A and B) ex. Chances of snow thurs. or Friday?

What are some features to a normal distribution?

-commonly observed shape of the probability distribution of continuous variables -Unbiased, continuous data "Bell curve" or gaussian distribution Ex. height, birthweight, test scores, highway speeds

What do need to help find binomial likelihood of success in N trials?

-number of successes were interested in -number of failures -success probability

the probabilities from all elementary event equals to ____

1

I polled students in my classes on their internet usage. M= 9.061 hours, SD = 6.319. What if I had a class that reported 15 hours online? What would this be in z- scores?

15-9.061/ 6.319 = z- score is 0.940 Use table to find # 0.8264 82.64% of students spend less than 15 hour online This class that spends 15 hours online is on average 83rd percentile - they use more internet than 83% of students

When the z-score is 2, what does it mean? When the z-score is -1.8, what does that mean?

2 SD above the mean -1.6 SD below the mean

If event A includes outcomes {1, 2, 3, 4, 5} and event B includes outcomes {4, 5, 6, 7, 8}, which outcomes are possible in A AND B?

4,5 because they occur in both sample space

The mean of SAT verbal score is 500 and SD 100. What is the probability that a test score is between 400-600?

400-500/ 100 = -1 600-500/ 100 = 1 Use table to find # 0.84134- 0.15866 = 0.6826 or 68.26% between 400-600

Standard Normal Table

Displays the area under the standard normal curve to the left of a z-score

True or False: You are in a casino, playing cards or the slot machine. It's been 30 mins and you've gotten only bad cards and low numbers, so it must be time to get high numbers.

False. this Idea is called gambler's fallacy. it's the mistaken idea that a given outcome can be determined from a previous patter of outcomes of independent events. they are independent, so whatever you got in the past does not affect what your next event will be

What is the probability that a random student uses the internet less than 13 hours a week? (i.e., what percentile would a student fall at if they used the internet 13 hours a week?) The mean is 9.06 hours and the SD is 6.32 hours.

First we calculate the z-score for this, which ends up being ~0.623. We find this value in the table and get p(<.62) = 0.73237, or the student is around the 73rd percentile of internet users

compound independent events

P (A or B): P(A) + P(B) - P(A) x P(B) Double counting so we much minus the intersection

What is the probability of my rolling a 5 on my die?

P(5)= 1/6

compound independent events: intersections

P(A and B) = P(A) x P(B) Ex. What's the probability of getting tails and 3 when rolling on die and flipping one coin? P (tails and 3) = ½ x ⅙ = 1/12

Probability of A and B

P(A) x P(B) ex. chances of snow Tuesday and Wednesday?

I am rolling a fair, six-sided die one time. What is the sample space S of this experiment?

S = {1, 2,3,4,5,6}

We have a sample space S of all whole numbers from 1 through 20. What is P(number greater than 14)?

There are only 6 numbers in the sample space higher than 14: {15, 16, 17, 18, 19, 20}, so the probability of getting one of these is 6/20 (or 3/10).

standard normal distribution

A normal distribution with a mean of 0 and a standard deviation of 1. Total area under curve = 1. -can be used to estimate probabilities

We have a sample space S of all whole numbers from 1 through 20. What is P(even number)?

Half of the numbers are even, so your probability of picking an even number is 0.5 (or 50%, or 1/2, or 1 out of 2).

We have a sample space S of all whole numbers from 1 through 20. What is P(even number OR greater than 14)? (You can assume these events are independent from one another).

Here, I would recommend using the formula over counting. P(A OR B) = P(A) + P(B) - P(A AND B) = 1/2 + 3/10 - 3/20 = 13/20. (Or, if you count: {2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19, 20}.)

What is the probability that a random student got between a 550 and a 650 on the SAT Verbal? (Mean = 500, SD = 100). Use that standard normal table to your advantage!

Here, we want p(550 < x < 650), but to do that we need the z-values of each. So 550 is 0.5 in z-scores and 650 is 1.5 in z-scores. So, p(0.5 < z < 1.5) = p(<1.5) - p(<0.5) = 0.93319 - 0.69146. This comes out to 0.24173 or around 24%.

What is the probability of scores being less than -1.15 on a standard normal curve?

I found the row with -1.1, looked at the column which corresponded to 0.05, and found the value which was 0.12507, or 12.507% of being less than -1.15.

What is the probability of scores being greater than 0.48 on the standard normal curve?

I want to find the value where the column is 0.4 and the row is 0.08. There, the value is 0.68439. However, I'm interested in what is GREATER than that, so I need to do 1-0.68439, which is 0.35161

What are some key parts to calculating probability for a normal distribution?

Use a solid line for normal distribution b/c data is continuous Y axis is probability density not just probability Total area under curve is equal to 1 Higher areas in the curve are more likely Need to find area under curve for a rough estimate (cumulative probability) the data is continuous so it is near impossible for a specific value to occur and we must look at a range of values

Using our z-score transformation formula, what is the equivalent z-score of an SAT score of 675, when the mean is 500 and the SD is 100?

We are looking for the z-score here, not the probability! So all we have to do is take the (value - mean)/SD = (675 - 500)/100. This gives us a z-score of 1.75

What is the probability of getting a value between 2.50 and 3.50?

We find this by getting the value of 3.50 and subtracting the value at 2.50. This is 0.99977-0.99379 which equals 0.00598.

We have a sample space S of all whole numbers from 1 through 20. What is P(even number AND greater than 14)? (You can assume these events are independent from one another).

You can solve this in two ways. One is that you can count the number of even numbers over 14: {16, 18, 20}, or you can use our formula: P(A AND B) = P(A) x P(B) = 1/2 x 6/20 = 6/40 = 3/20.

What does probability distributions help address?

addresses "is this real?" question in our statistical questions -leads into inferential stats -whether likelihood of event is real or not

Anything that happens (outcome) is _____

an event (E or X)

Let's say you have discrete data and the scale of measurement is categorical, which probability distribution can be used?

binomial (2 variables) chi square (2+ variables)

What are the different types of probability distributions?

binomial, normal, t, chi square, and F distribution

Changing the mean and variance by adding and multiplying by constants is called _____

data transformation ex. Fahrenheit to celsius or inches to cm

the occurrence of set of occurences that events happen with some probability is called _____

defined experiment