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, none of the 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, wrongly sometimes.)
Independence: Read Ash pp
41-43. You can skip "Method 2" and Prob of A before B.
Read M&M pp. 294-296, 340 Rule 5, 350.
(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, in class)
(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.
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 (2nd and 3rd can be anything. Is it
the same answer for W on the 1st and the 2nd? See "time
blindness" above. If you don't buy the time blindness, do a tree
and calculate the answer from it.)
M&M p. 300 ff: 4.22,
4.30, 4.31, 4.32, 4.33, 4.34, 4.35 (Some may be repeats from last
semester.)
What's Next? , review M&M pp. 305-309 (Random variables), pp 366-371, 380-82 (Binomial) 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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