STATS Unit 6
Geometric model on calculator
2nd -> VARS -> geompdf and 2nd -> VARS -> geomcdf use pdf when going to an exact number (ex. probability that the 5th person has an item, prob of 5th trial being a success) use cdf when trying to be within a number (probability that you would find an item in 6 OR FEWER trials, probability of a success WITHIIN 6 (6 or fewer) trials)
What is the difference between a binomial and geometric (bernoulli trial) model problem?
Binomial - fixed number of trials, counting successes - keeps counting after one success Geometric - counting how many trials until success, STOPS COUNTING after a success
The mean weight of a male llama is 335 pounds with a standard deviation of 27 pounds. The mean weight of a female llama is 287 pounds with a standard deviation of 24 pounds. If it's normally distributed, how would you find the probability of the female llama weighing more than the male?
Find the expected value E(F-M) ^female mean weight minus male mean weight Find the standard deviation SD(F+M) draw a bell curve and do a normal model problem with the lower bound being the cut point (z score) at 0 and the upper bound being 99 This gives you the probability of the difference between the female and male weight being higher than 0 - in other words probability of female weighing more than male
Bernoulli Trial
a problem that satisfies the following conditions: 1) there are two possible outcomes (success/fail) 2) the probability of success, p, is constant 3) the trials are independent - sometimes meaning they satisfy the 10% condition (as long as a sample is less than 10% of a population, we say it's independent) ex) drawing skittles from a bag, what's the probability that the first speckled one is the 4th candy we get?
Random Variable (how is it denoted)
a variable based on the outcome of a random event. denoted with capital X ex) the amount of money you win from a slot machine game
adding or subtracting two random variables ______ their means
adding or subtracting 2 random variables adds or subtracts their means
adding or subtracting multiple random variables causes their variance to _________ variance is _________ and always __________`
adding/subtracting multiple random variables causes their variance to add variance is distance, always positive so, if E(X+Y) = E(X)+E(Y) If x and Y are independent, Var(X+Y) = Var(X) + Var(Y) and standard deviation is the square root of the resulting variance above
10% rule
bernoulli trial problems MUST have independence to be bernoulli trial problems 10% rule - exception to independence. If a sample is less than 10% of the population we say it's independent.
How would you find the probability of getting AT LEAST a certain number of successes in a binomial model problem?
binomial cdf finds the probability of getting AT MOST a certain number of successes (ex. 3 or fewer) so to find prob of AT LEAST a certain number of successes: take complement of binomcdf, and subtract 1 from x ex) probability of getting at least 2 speckled skittles out of five, probability of speckled skittles = .3 1-binomcdf(5,0.3,1) ^ see how x would normally be 2 here, but we subtracted 1
What can be said about the standard deviations in the following problems: 1) A printer has a mean cost of $30 with a standard deviation of $5, a computer has a mean cost of $100 with a standard deviation of $3, what is the standard deviation of a computer and a printer (bought together) 2) Which would cost more on average - the printer or computer - and with what standard deviation
both problems would have the same standard deviation even though the second problem would require subtracting the means the stdev is the same as the first problem since stdev can only be added
Continuous variables ______(can/cannot) be added and subtracted. When two independent continuous random variables have normal models, so does their _________
can combination
Adding and subtracting constant data affects the ________ of the data but not the _________
center, spread
Geometric probability model (what does it do)
finds probability of a random variable by counting bernoulli trials until the first success
multiplying standard deviations (is/is not) the same as adding them
is not the same when multiplying standard deviations, everything in the probability model is multiplied by a factor- for example, if a casino decided to have a special night where all the possible winnings on a lottery game were doubled (everything mult by 2) when adding standard deviation you would have to find the square root of the sum of the variances (variance = stdev^2). For example, finding the standard deviation of expected winnings from playing on 2 slot machines (the values stay the same, but there are 2 random events, so you add)
What is n, p, and x in binomial model problems?
n - number of trials p - probability of success x - number of successes
How do you find the expected value in a geometric probability (bernoulli trial) problem?
often these are problems that ask us to count trials UNTIL the first success, or the first occurrence of something E(X) = μ = 1/p where p = probability of success ex) if the probability of drawing a speckled skittle from a bag is .3, how many times do we have to check on average to find a speckled one? E(x) = μ = 1/p = 1/.3 = 3.33 times
Suppose X and Y are independent random variables. The variance of X is equal to 16 and the variance of Y is 9, let Z be X-Y how do you find the standard deviation of Z?
stdevZ = sqrt(16+9) = 5
Binomial model (what does it do?)
tells us the probability for a random variable that counts the number of successes in a fixed number of trials
probability model, net winnings
the collection of all possible values and probabilities that can occur ex) see picture, usually they'll have the outcomes, the net winnings from those outcomes (if it's a game to get money) and the probability net winnings means the amount of money earned minus the amount of money it costs to play
variance (how do you find it?) and standard deviation (what does it describe?)
the expected value of squared deviations from the mean to find - use the calculator or square the standard deviation standard deviation - square root of the variance. describes the spread in the model.
Expected value
theoretical long run average of a random variable formula - E(x) = ∑x(P(x)) ^ ex) each amount of money won from slot machine times probability of getting that amount, so if you can win $25 with a probability of .25, $50 with a probability of .15, and $100 with a probability of .10, it would be: E(x) = 25(.25)+50(.15)+100(.10)
On valentine's day at charlotte's cheese shack couples get a discount of either $10 or $20 with a 50/50 chance. When two couples dine together on a single check Charlotte doubles the discount offer they receive. how would you find the expected value and standard deviation of the expected discount when couples dine together?
to find the standard deviation and expected discount STAT -> Edit -> enter amount discount in L1 and probability in L2 STAT -> CALC -> 1 VAR statistics, set List to L1 and FreqList to L2, calculate It should show you the expected value (x bar) and the standard deviation (σx) multiply that expected value and standard deviation by 2 since the discount is doubled
Discrete variable (give example)
variable that has specific values and that cannot have values between these specific values ex) if a slot machine offers a $25, $50 or $100 reward then 1 - the amount of money you win would be a random variable (based on random event) 2 - The amount of money is also a discrete variable because it's limited to a few options ($25, $50, $100)
How do you solve continuous random variable problems?
-if the problem requires adding/subtracting means or adding standard deviations, do that first -now that you've found the mean and standard deviation, draw a bell curve for it -mark the mean and standard deviation and the cut points -find the z score for the cut points of the upper and lower bound z=(x-x̄)/stdev -plug into 2nd -> vars -> normalcdf upper, lower BASICALLY solve it like a normal model problem, just make sure to add and subtract means/add standard deviations if the problem requires it
How/when do you use the inverse of geometric/binomial probability?
-probability that something WILL NOT happen ex) probability you won't find a success before the 10th trial - 1-geomcdf(p,10) p=probability ex) probability you won't get 2 speckled skittles out of 5 drawn, p=.3 1-binompdf(5,.3,2)
When you add or subtract multiple random variables how do you find the standard deviation?
-square the standard deviation of the two random variables to find the variance -add the variances together -take the square root of the result ex) (stdev of x)^2 = Var(x) (stdev of y)^2 = Var(y) If x and Y are independent, Var(x) + Var(y) = Var(x+y) and standard deviation is the square root of the resulting variance above
What are the 3 requirements on the AP Rubric for probability problems??
1) name the distribution with words or a proper formula (possible we haven't learned this yet!) 2) identify correct parameters (use standard notation like n=1, p=.7, μ = 65) 3) correct calculations
Binomial model on calculator
2nd -> VARS -> binomcdf and 2nd -> VARS -> binomcdf pdf is for exact numbers (probability of getting 2 successes in 5 trials) cdf is when you see a keyword like fewer or smaller (probability of getting 2 OR FEWER successes in 5 trials)
On valentine's day at charlotte's cheese shack couples get a discount of either $10 or $20 with a 50/50 chance. When two couples dine together on a single check Charlotte doubles the discount offer they receive. what would the expected value and standard deviation be if two couples dined together on a separate bill?
If two couples dined together, the expected value would be multiplied by two and you would ADD the standard deviation of the couples discounts (don't multiply by 2) remember to add stdev you have to add the variances then take the square root don't add stdev directly you only multiply standard deviation if all the values in a probability model are being multiplied by two - like when she doubled the discount - but when two couples dine together you add the stdev since those are 2 separate instances of the probability. (adding stdev is not the same as multiplying it)
In a geometric probability model, p is ______ , q is ______, and x is ______
p = probability of success q = probability of failure (previously called complement, 1-p) x = number of trials until the first success
The standard deviation for the number of repairs a computer system needs per year is .5 If a company has a lease on the computer for 3 years, what is the standard deviation for the number of repairs the computer would need at the end of the 3 year period?
sqrt(.5^2+.5^2+.5^2) you add the variances and square root it you wouldn't multiply by 3 because you only do that when all the values in a probability model are multiplied by a common factor - but in this case we're dealing with 3 separate random events (3 years and the # of repairs each year) so we add the variances
If wins at a lottery machine have a standard deviation of 120, what is the standard deviation of the casino's payout if gamblers play this machine 10000 times a day?
sqrt(120^2*1000)
Standard deviation is always _________ because we can't have a negative mean
stdev is always an addition problem - always add standard deviations of random events, even if the means are being subtracted.
When we multiply, the mean and standard deviation are _____________, but the variance is is ___________
when we multiply the mean and stdev are multiplied by that number, but the variance is multiplied by that number squared.