CSE390D Notes for Friday, 11/15/24
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more terminology:
probability distribution: assignment of probabilities to all
elements of sample space (finite set)
uniform distribution: all outcomes equally likely, p = 1/n
0 <= p(s) <= 1
sum(p(s)) for s in S = 1
sum rule:
P(E) = sum(P(s)), s in E
still have:
P(E') = 1 - P(E)
conditional probability:
P(E | F) = P(E intersect F) / P(F)
S = flip a coin 5 times
F = first coin is a tail
E = at least 3 heads
P(F) = 2^4/2^5 = 1/2
to find P(E intersect f):
starts with T
then at least 3 H
we can enumerate: THHHH, TTHHH, THTHH, THHTH, THHHT
P(E intersect F) = 5/2^5
P(E | F)? = 5/2^5 / 1/2 = 5/2^4 = 5/16
independence
E and F are independent iff P(E intersect F) = p(E) * p(F)
P(E | F) = P(E intersect F)/P(F) = P(E) * P(F)/P(F) = P(E)
corollary: E and F independent iff P(E | F) = P(E)
independence quiz:
flip a coin 3 times
E: first coin is a head (HHH, HHT, HTH, HTT)
F: second coin is a head (HHH, HHT, THH, THT)
E intersect F: (HHH, HHT)
P(E) = 4/8 = 1/2
P(F) = 4/8 = 1/2
P(E intsersect F) = 2/8 = 1/4 = 1/2 * 1/2 = P(E) * P(F)
yes
roll two dice:
E: sum of dice is 5 [(1, 4), (2, 3), (3, 2), (4, 1)]
F: first die is 1 [(1, 1), (1, 2), (1, 3), (1, 4), (1, 5),
(1, 6)]
E intersect F: [(1, 4)]
P(E) = 4/36 = 1/9
P(F) = 6/36 = 1/6
P(E intersect F) = 1/36 != 1/9 * 1/6 = P(E) * P(F)
no
roll two dice
E: sum is 7 [(1, 6), (2, 5), (3, 4), (4, 3), (5, 2),
(6, 1)]
F: fist die is 1 [(1, 1), (1, 2), (1, 3), (1, 4), (1, 5),
(1, 6)]
E intersect F: [(1, 6)]
P(E intersect F) = 1/36 = 1/6 * 1/6 = P(E) * P(F)
yes
family has 3 children
E: children of both sexes (BBG, BGB, GBB, GGB, GBG, BGG)
F: at most one boy (GGG, BGG, GBG, GGB)
E intersect F: (BGG, GBG, GGB)
P(E) = 6/8 = 3/4
P(F) = 4/8 = 1/2
P(E intersect F) = 3/8 = 3/4 * 1/2 = P(E) * P(F)
yes
Bernoulli trials
success probability p, failure probability q (p + q = 1)
P(exactly k successes in n independent trials) =
(n choose k) * p^k * q(n - k)
example:
A coin is biased so that the probability of heads is 2/3. What is
the probability that exactly four heads come up when the coin is
flipped seven times, assuming that the flips are independent?
(7 choose 4) * (2/3)^4 * (1/3)^3
random variable
A random variable is a function from sample space to real numbers
The distribution of a random variable X on a sample space S is the
set of pairs (r, P(X=r)) for all r in X(S) where P(X=r) is the
probability that X takes the value r.
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Stuart Reges
Last modified: Fri Nov 15 15:25:38 PST 2024