Recent content by tmt1

Given $P(PH | H) = 0.8$ and $P(PH | \lnot H) = 0.3 $ and $P(H) = 0.1$ how can I derive $P(PH)$ without resorting to a joint distribution table?

Given $$P(B \land C)$$ will it always be true that $$P(B \land C | A) P(A) + P(B \land C | \lnot A) P( \lnot A)$$ (regardless what $P(A)$ would be)? How can I prove this?

Given this base data (taken from Graphical Models )$P(C) = 0.5$ $P(\lnot C) = 0.5$ $P(R | C) = 0.8$ $P(R | \lnot C) = 0.2$ $P(\lnot R | C) = 0.2$ $P(\lnot R | \lnot C) = 0.8$ $P(S | C) = 0.1$ $P(S | \lnot C) = 0.5$ $P( \lnot S | \lnot C) = 0.5$ $P( \lnot S | C) = 0.9$ $P(W | \lnot S, \lnot...

Oh yeah, finally clicked. Simple a matter of P(A | B) = P(A ^ B) / P(B)

No -- there are 25 reservations every day. And there is a 0.8 probability that a reservation is claimed. Therefore, each day it is expected that 20 cars will be claimed (25 * 0.8). The probability that 23 or more of reservations are claimed ${{25}\choose{23}} {0.8 ^{23} * 0.2^ 2}$ +...

So, at a car rental company, 20% of car reservations are not claimed. There is a total of 22 cars and the manager takes 25 reservations a day. If all cars are claimed for a day, what is the probability that one or more customer who had reservations were unable to claim their car? I need to...

I'm getting these concepts confused. If I have an object called $x$, and I have five places or slots to put the object, how many ways could 2 $x$s be places in the 5 spaces? Example: x x _ _ _ x _ x _ _ x _ _ x _ x _ _ _ x _ x x _ _ _ x _ x _ _ x _ _ x _ _ x x _ _ _ x _ x _ _ _ x x So in...

I have $P(B) = 0.4$ and $P(\lnot B) = 0.6$. $P(TS|B) = 0.7$ and $P(TS|\lnot B) = 0.25$ $P(B|TS) = 0.65116$ and $P(\lnot B|TS) = 0.34884$ (from bayes theorem). Now, if we get $B$ or $\lnot B$, and we get the same event twice in a row so we get $B$ then $B$ or $\lnot B$ then $\lnot B$, what...

How to translate "there exists exactly one happy person" into predicate logic? I came up with $$ \exists x : happy(x) \implies \forall y: happy(y) \land y = x$$. But this is incorrect. I also tried $$\exists x: happy(x) \land \forall y: happy(y) \land x = y$$. This is also incorrect. The...

A coin is tossed 4 times. Is there a way to determine mathematically what is the probability that exactly 2 heads occur? By drawing a decision tree I can determine that it is 6/16, but this seems like an arduous process for larger numbers.

Is $$x^\frac{2}{3} (\frac{5}{2} - x)$$ a continuous function for all values of x? It seems disjointed at $x = 0$ but the limit as x approaches 0 is 0 from both sides of x.

$$\int_{}^{} \frac{1}{x\sqrt{x^2 + 16}} \,dx$$ I can set $x = 4 tan\theta$. Thus $dx = 4 sec^2 \theta d\theta$ So, plug this into the first equation: $$\int_{}^{} \frac{4 sec^2 \theta }{4 tan\theta \sqrt{16 tan^2\theta + 16}} \,d\theta$$ Then, $$\int_{}^{} \frac{ sec^2 \theta }{4 tan\theta...

If anyone value in an expression is approaching 0, does the entire expression equal 0? So for example, for the limit $\lim_{{z}\to{q}} z ln(z) f(z)$. If $\lim_{{z}\to{q}}f(z) = 0$, then does $\lim_{{z}\to{q}} z ln(z) f(z)$ equal 0?

I have: $$\int_{1}^{3} \frac{1}{\sqrt{3 - x}} \,dx$$ I can do $u = \sqrt{3 -x}$, and $du = \frac{1}{2 \sqrt{3 - x}} dx $ and $dx = 2 \sqrt{3 - x} du $. Plug into original equation: $$\int_{1}^{3} \frac{2 u }{u} \,du$$ and $2 \int_{1}^{3} \,du = 2u = 2 \sqrt{3 - x} + C$ So $(2\sqrt{3 - 3})...

I have: $$\int_{1}^{2} \frac{1}{x lnx} \,dx$$ I can set $u = lnx$, therefore $du = \frac{1}{x} dx$ and $xdu = dx$. Plug that into the original equation: $$\int_{1}^{2} \frac{x}{x u} \,du$$ Or $$\int_{1}^{2} \frac{1}{ u} \,du$$ Therefore: $ln |u | + C$ and $ln |lnx | + C$ So I need to...