Module 7 part 2
Dependent Events - conditional probability example
14 marbles left since first marble was not replaced 2nd problem only 5 green marbles left since 1st green marble was not replaced
Determining Independence - 3rd Formula
3rd formula is really the same as the first two. General multiplication formula which say's that the probability of A and B is equal to the probability of A times the probability of B given A. A and B will be independent if this condition does not matter. We can ignore this condition here and just multiply the two individual probabilities together to get the "and" probability.
Determining Independence Example - not A
B is the event of selecting an orange toy and A is the event of selecting a truck, the event B given A represents the event that we selected an orange toy if we know we got a truck, since getting a truck is our condition or our given in this conditional probability 1 out of 1 truck is orange 7 toys are not trucks, none are orange not equal - dependent
Determining Independence - 4th Formula
Boils down to knowing if the condition in a conditional probability matters. If the condition does not matter, then these formulas will hold and the events will be independent. If the condition does matter, that means that the events by definition are dependent and these formulas will not hold.
"And" and "Or" probability formulas
Disjoint Events - A and B = 0 since they can not happen at the same time - no overlap that will be counted twice. "Or" probability - add, subtract overlap if not disjoint "And" probability - multiply
Determining Independence - independent
Probability of B = 4/8 (size of sample space is 8, 4 out of 8 toys are cars) Probability of B given A = 1/2 (B is selecting a car, A is selecting a green toy, sample space is 2 since there are 2 green toy, 1 out of 2 green toys is a car); green toy is a given (the condition), what is the likelihood it is a car? They were both equal to one-half. So that tells us that this condition did not matter. Knowing that we selected a green toy, did not affect our chances of selecting a car. And since the condition didn't matter, that means that these events are *independent*.
Determining Independence Example - dependent
Probability of selecting a red toy *and* a doll *and* use multiplication formula 2 out of 8 toys are red and 1 out of 8 toys are a doll Probabilities is *not* equal, events are *dependent*
Probability Tree
Probability of what colored socks are pulled out in what order, socks not replaced after pulling out reducing size of sample space by 1 each time. "OR" - add
A and B not disjoint and not independent
marble not replaced 1st problem - 14 marbles left after drawing blue marble 2nd problem - 14 marbles and 5 green left after drawing a green marble
Formula for *Independent Events*
if A and B are independent events, then the probability of A and B is equal to the probability of A times the probability of B
Determining Independence - 1st and 2nd Formula
First Formula Two events are independent if the outcome of one has no effect on the occurrence of the other. So if A and B are independent, then knowing that B already happened should have no effect on the probability of A. When events are independent, the condition won't matter because it doesn't affect anything. We can compare the conditional probability with the regular probability without this condition to see if the condition makes a difference. If the probability of A given B is equal to the probability of A without the condition, that means that knowing that B already happened had no effect on the probability of A. So that means that these events are independent. If these two probabilities are not equal to each other, that means the events are dependent, knowing that B happened did have an effect on the probability of A. Same reasoning works on 2nd formula in list.
Determining Independence using Frequency Tables
Having a relapse and having a relapse while taking desipramine Patient taking desipramine has a much lower chance of having a relapse than all those in the study group Condition of desipramine affected the probability, events are *dependent*
Probability Tree
I - infected (1/200=0.005) not I - not infected (199/200=0.995) Po - positive (infected and test Po - 0.80) N - negative (infected and test N - 0.20) not I and test Po - 0.05 not I and test N - 0.95 Probability of I and test Po or not I and test Po *OR* add together
Probability Tree
Person is infected and test negative
Determining Independence Example - dependent
We know we selected a car, what is the likelihood it was pink? 1 of 8 toys are pink, 1 of 4 cars are pink. Probabilities is *not* equal, events are *dependent*
Determining Independence
What we are taking a percentage of? Is it a percentage of the whole sample space or is it a percentage of a smaller group? That's what makes this third one a conditional probability. Conditional probability - the probability of earning a Master's degree if we're only looking at the females.
A and B not disjoint and not independent
additional rule is same multiplication - top formula if A is the condition, bottom formula if B is the condition Always divide by the condition
Dependent Events - conditional probability example
card not replaced in deck, 51 cards left 1st problem - diamond is red, only 25 red cards left
Frequency Table Conditional Probability
only looking at patients with no relapse in a conditional probability that we are finding the probability of something if we already know something about the situation, which is our condition or our given
independent formula
overlap area counted in twice so it must be subtracted once this formula works for any two events A and B, independent, dependent, disjoint, or not disjoint. If A and B happen to be disjoint, then last part of formula will be zero since disjoint means that there is no overlap.
Independent/Dependent Events
two events are disjoint if they cannot occur at the same time events will be dependent if the occurrence of one does affect the probability that the other will occur one event is that a person develops lung cancer and the other event is that a person is a smoker, these events will be dependent If two events are disjoint, we know that they cannot happen at the same time. So if one of those events occurs, that automatically makes the probability of the other event occurring zero because they cannot both occur at the same time. Since the occurrence of the first event has affected the probability of the second event, it made that probability zero, these events are dependent. Since the occurrence of the first event has affected the probability of the second event, it made that probability zero, these events are dependent.
