B. Today a small grocery store has 6 cartons of
milk, 2 of which are sour.
a) If you are going to buy the 4th carton of
milk sold today at random, compute the probability that your carton is
sour. Do it by making a tree; first carton sold Sour or Not, 2nd
Sour or Not, etc. up to your 4th.
b) If you buy the first carton of milk today,
what is the probability that it is sour?
(Note, neither of these final probabilities are conditioned on the past. So my getting a heart IF person one or two already has, does depend on those previous events. But my getting a heart considered by itself alone, does not. Our intuition tends to "know" that the past matters.)
We'll start here Wednesday
Independence: Read Ash pp
41-43. You can skip "Method 2" and Prob of A before B.
Read M&M 301-305, 356-57
(1) Definition: A and B are independent
iff P(A and B) = P(A)·P(B)
Intuitive:
Event A and event B are independent if the occurrence or nonoccurence of
one makes no difference to the probability of the other.
(2) Formula: P(B|A)
= P(B)
(1) and (2) are (almost) equivalent, since
P(B|A) = P(BandA)/P(A). (proofs)
(3) By symmetry, then, P(A|B) = P(A).
From the intuitive idea, it seems clear that
if A and B are independent then so are Acand B,
A and Bc , and Acand Bc
(Proofs: Handout)
Then P(B|Ac) = P(B) = P(B|A), the
rest of the intuitive idea.
For constructing models: we think not
just of events being independent, but whole mechanisms or processes.
Every event in process A being independent of any event in process B.
(e.g. Flip a coin, then draw a card from a deck)
Trees: If two processes are independent,
and can be thought of as sequential in a tree, all the branchings at the
second stage will be identical, whatever the preceding result.
Homework: A and B from top of page.
(And finish Mini-exam)
Postpone to Wednesday:
Handout--show: If A and B are independent,
then so are Acand B, A and Bc ,
and Acand Bc
Ash p. 45: 8, 11. 12 (is it
the same for W on the 1st and the 2nd? See "time blindness" above)
M&M p. 310 ff: 4.28, 29,
30, 32, 34, 36 (Some may be repeats from last semester. Solutions
are in the Math Clinic)
What's Next? , review M&M pp. 312-317 (Random variables), Ash sec. 2.2 Multinomial and Binomial distributions. (Ash does Multinomial first, and Binomial as a special case. This approach doesn't "privilege" the binomial "successes" as much as M&M's approach did.)
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