2009 Lista de Exercícios (Conceitos Básicos de Pr

Universidade Federal de Juiz de Fora
Mestrado em Modelagem Computacional – 2009
Lista de Exercícios (Conceitos Básicos de Probabilidade)
1) Seja um sistema em que dois tipos de mensagens possam ser enviadas do nó A para o nó B.
A probabilidade de envio de uma mensagem do tipo M1 é de 0.6 e do tipo M2 é 0.4. Essas
mensagens podem chegar com dois tipos de erro no destino (E1 ou E2). A probabilidade da
mensagem M1 chegar no destino com erro E1 é 0.05 e com o erro E2 é 0.03. A
probabilidade da mensagem M2 chegar no destino com o erro E1 0.01 e com o erro E2 é
0.05. Calcule a probabilidade de uma mensagem chegar errada no destino.
2) A group of four VLSI chips consists of two good chips, labeled g1 and g2, and two
defective chips, labeled d1 and d2 If three chips are selected at random from this group,
what is the probability of the event E = “two of the three selected chips are defective?”
3) We are given a box containing 5000 VLSI chips, 1000 of wich are manufacturated by
company x and the rest by company Y. Ten percent of the chips made by company X are
defective and 5% of the chips made by company Y are defective. If a randomly chosen chip
is found to be defective, find the probability that it came from company X.
4) A microcomputer system consists of a microprocessor CPU chip and a random access main
memory chip. The CPU is selected from a lot of 100, 10 of wich are defective, and the
memory chip is selected from a lot of 300, 15 of which are defective. Define A to be the
event “the selected CPU is defective”, and let B be the event “the selected memory chip is
defective”.
5) Consider the experiment of rolling two dice. Let the sample sampe S = {(i,j) | 1 <= i,j <= 6.
Also assume that each sample point is assigned a probability of 1/36. Define the events A,B
and C so that A = “first die results in a 1,2, or 3”; B = “first die results in a 3,4, or 5”; C =
“the sum of the two faces is 9.” What is the probability of: (A∩B), (A∩C),(B∩C) and
(A∩B∩C) ?
6) A given lot of VLSI chips contains 2% defective chips. Each chip is tested before delivery.
The tester itself is not totally reliable so that
P (“tester says chip is good”| “chip is actually good”) = 0.95
P(“tester says chip is defective”| “chip is actually defective”) = 0.94
If a tested device is indicated to be defective, what is the probability that it is actually defective?
7) A manufacturer produces VLSI chips, 1% of which are defective. Find the proability that in
a box containing 100 chips, no defectives are found.
8) Of all graduate students in a university, 70% are women and 30% are men. Suppose that
20% and 25% of the female and male population, respectively, smoke cigarettes. What is the
probability that a randomly selected graduate student is:
(a) A woman who smokes? (b) A man who smokes? c) A smoker?
9) Suponha que A e B sejam eventos mutuamente exclusivos, e que P(A) = 0,3 e P(B) = 0,5.
Qual a probabilidade que:
a) Apenas A ou B ocorram.
b) A ou B não ocorram.
10) The interarrival time for packets passing through a switch are r.v.s with the following
density function:
f_T(t) = e^{-t}; t >=0
= 0
;t<0
where the interarrival times are measured in ms. The sucessive experimental values of the
durations of these interarrival times are recorded in records with 4 entries. As soon as a
record is filled it is written to disk and a new record is started.
(a) Determine the mean of T
(b) Given that no message has arrived in the last 2ms, determine the distribution of K, the
number of messages that arrive in the next 8ms
11) Let the discrete random variable N_t denote the number of jobs arriving to a file server in
the interval (0,t]. Let X be the time to the next arrival. Further assume that N_t is Poisson
distributed with parameter λt , so that λ is the arrival rate. What is the probability of the
next arrival be greater than t? Hint: considerer that no arrival occurs up to time t.