Chapter 9 Section 6: Properties of Probability

Hey everybody! Since I'm posting this blog before Mr. French has lectured on the section, please excuse any mistakes. I will be sure to correct anything that is wrong as soon as I'm aware of it.

Intersection of Events

AND:

1.) p(A and B)=p(A) x p(B) Independent events
2.) p(A and B)=p(A) x p (B/A) Non-independent events
~Note: In the notation (B/A), the "/" symbol is not a division sign. (B/A) means "the number of ways B can occur given that A has occurred)

OR:

1.) p(A or B)=p(A) + p(B) Mutually exclusive: no overlap
2.) p(A or B)=p(A) + p(B) - p(A and B) Not mutually exclusive: overlap exists

Example 1:
a.) Sally has a crush on Billy and Fred. Each morning on her way into homeroom she passes the two boys: Billy is always at the lockers and Fred is always around the corner by the drinking fountain. The two boys can't see each other, so the probability of each one waving to Sally is not impacted by the other's actions. If there is a 50% chance that Billy will wave to Sally (he's sometimes in a sour mood) and there is an 80% chance that Fred will wave to Sally, what is the probability that tomorrow morning, both boys will wave to Sally?

Solution: Since the boys' actions are not impacted by one another, this is an independent event, and thus we have to follow the #1 equation under AND. All we do is plug in: p(B) x p(F)=.5 x .8=.4 >>> 40%

b.) Fred began to realize that the object of his affection was toying with his emotions, so now he is spying on Sally. If Billy waves to Sally at the lockers, there is only a 20% chance that Fred will then wave to her at the drinking fountain. If Fred catches Billy in the act, what is the probability that he will take pity on silly Sally and wave to her?

Solution: Because p(F) is affected by p(B), we now have to use equation #2. Since the question already gave us the probability of F given B, we plug in: p(B) x p(F)=.5 x .2=.1 >>> 10%

Complementary Events

p(A) + p(not A)=1

~ Note: This equation is true because in a given situation, the possibility of "A" and the possibility of "not A" are the only options available, and thus when added, they equal 1.

Example 2:

Probability that Billy waves: 50%

Probability that Fred waves: 80%

a.) What is the probability that Fred does not wave?

Solution: Given the equation above, we switch it around and plug in our value of 80%: 1-.8=.2 >>> 20%

b.) What is the probability that only one of the boys waves?

Solution: In this situation, we should first calculate the two individual possibilities (Billy waves and Fred does not wave or Billy does not wave and Fred waves).

Billy waves and Fred does not wave: .5 x .2=.1 >>> 10%

Billy does not wave and Fred waves: .5 x .8=.4 >>> 40%

Now we add the two probabilities together: 10% + 40%=50%

Additional Links: http://www.mathepi.com/stat/prob/compprob.html http://regentsprep.org/Regents/math/andor/TAndOr.htm

Lucy-you're next!!

This is a poem that I read last year in Ms. Yelverton's class. It's in a huge anthology of modern poetry, which I took off my shelf this weekend so that my sister could borrow it for her poetry project, and I "wasted" about 30 minutes of valuable SAT time going back through the poem. This is one of my favorites.

Having a Coke With You

is even more fun than going to San Sebastian, Irun, Hendaye,

Biarritz, Bayonne

or being sick to my stomach on the Travesera de Gracia in

Barcelona

partly because in your orange shirt you look like a better

happier St. Sebastian

partly because of my love for you, partly because of your love

for yoghurt

partly because of the fluorescent orange tulips around the

birches

partly because of the secrecy our smiles take on before people

and statuary

it is hard to believe when I'm with you that there can be

anything as still

as solemn as unpleasantly definitive as statuary when right in

front of it

in the warm New York 4 o'clock light we are drifting back and

forth

between each other like a tree breathing through its

spectacles

and the portrait show seems to have no faces in it at all, just

paint

you suddenly wonder why in the world anyone ever did them

I look

at you and I would rather look at you than all the portraits in the

world

except possibly for the Polish Rider occasionally and anyway it's

in the Frick

which thank heavens you haven't gone to yet so we can go

together the first time

and the fact that you move so beautifully more or less takes

care of Futurism

just as at home I never think of the Nude Descending a Staircase or

at a rehearsal a single drawing of Leonardo or Michelangelo

that used to wow me

and what good does all the research of the Impressionists do

them

when they never got the right person to stand near the tree

when the sun sank

or for that matter Marino Marini when he didn't pick the rider

as carefully

as the horse

it seems they were all cheated of some

marvellous experience

which is not going to go wasted on me which is why I'm

telling you about it

-Frank O'Hara