Innehåll 1 Sannolikhetsteori - Matematikcentrum

1 SANNOLIKHETSTEORI
(a) If there are two trucks in the queue when a truck arrives at the quarry, what is the probability
that its “waiting time” will be less than 5 minutes
(b) Before arriving at the quarry (and thus not knowing the size of the queue), what is the probability
that the waiting time of a particular truck will be less than 5 minutes
11. [∗] Two cables are used to lift a load W (see figure). However,
normally only cable A will be carrying the load; cable B is slightly
longer than A, so normally it does not participate in carrying the
load. But if cable A breaks, then B will have to carry the full load,
until A is replaced. The probability that A will break is 0.02. The
probability that B will fail if it has to carry the load by itself is
0.30, but is 0 as long as A carries the load.
(a) What is the probability that both cables will fail
(b) If the load remains lifted, what is the probability that none of the cables have failed
12. [∗] Ett nytt test för att avslöja en allvarlig sjukdom har tagits fram. Det ger positivt utslag med
sannolikheten 0.99 om personen har sjukdomen fast med sannolikheten 0.05 även om personen
inte har den. Det anses vara känt att 1 % av patientmaterialet har sjukdomen.
(a) Beräkna den intressanta sannolikheten att en patient har sjukdomen om testet är positivt.
(b) Vilken egenskap hos testet ska man försöka ändra för att få en högre sannolikhet i (a) Ska
man försöka få 0.05 att bli 0 eller 0.99 att bli 1
(c) Antag att testet istället används i ett land där 50 % har sjukdomen. Vilket svar ger då frågan i
(a)
13. A contractor is submitting bids to 3 jobs, A, B and C. The probabilities that he will win the jobs
are P(A) = 0.5, P(B) = 0.8 and P(C) = 0.2, respectively. Assume events A, B, C are statistically
independent. Let X be the total number of jobs the contractor will win.
(a) What are the possible values of X Compute and plot the probability mass function (sannolikhetsfunktionen)
of the random variable X .
(b) Determine P(X ≤ 2).
(c) Determine P(0 < X ≤ 2).
14. Ett lokaltåg skall ankomma till en station kl 13.03 men brukar vara något försenat. Förseningen
(enhet: minut) varierar så att den kan betraktas som en s.v. X , som har täthetsfunktionen f X (x) =
1/5 om 0 ≤ x ≤ 5. Hur stor är sannolikheten att tåget kommer senare än 13.06 Hur stor är
sannolikheten att det kommer mellan 13.04 och 13.05
15. The settlement of a structure has the probability density
function (täthetsfunktion) shown in the figure.
(a) What is the probability that the settlement is less
than 2 cm
(b) What is the probability that the settlement is
between 2 and 4 cm
(c) If the settlement is observed to be more than
2 cm, what is the probability that it will be less
than 4 cm
f (x) X
h
0 2 4 6
x, settlement in cm
4