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