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Unformatted text preview: Midterm Examination of Probability Datci90f04s’23 l.(20%)Please preve the fellewing statements: (a)The prcbability cf selecting 11’2 in a tandem selectien at a paint ﬁem (0,1) is 0. (10%) (b) A paint is selected at randem frem interval (0,1). The prebability that is irratieual is 1. (10%) 2.(10%)If twc fair dice are rolled 10 times, what is the prebahility ef at least cne 6 (en either die) in exactly ﬁve cf these 10 rclls.‘ 3.(1 [1%)me families with three children, a famin is selected at randcm and feund te have a hey. What is the pfﬂbﬂbilily that the hey has a brether and a sister? Assume that in a three-chi 1d famin all gender distributicn have equal prcbabilities. 4.(l 0%)An urn ccntains live red and three blue chips. Suppcse that feur cf these chips are selected at randern and transferred to a seccnd urn, which was criginally empty. If a randcm chip selected ﬁcm this seccnd urn is red, what is the prcbability that twe red and twc blue chips were transferred frcm the ﬁrst urn te the seccnd urn? 5.(l [I%)1ﬂ S be the sample space cf a repeatable experiment. Let A and B be the mutually exclusive events cf S with a prebabilit},r cf PM) and 19(3), respectively. In independent trials cf this experiment, what is the prebability that the event A cccurs befcre the event B ‘? 6.(8 %)In this Prcblem, we censider that a randcm variable X has the distributicn functien F(t) given by 0 iii-l (lf4)t+lf4 -1 a: tee F(t)= 1x2 0 a tel (1f12)t+?f12 1 2 H2 1 2 a t. (a) Plet the graph cf Ht). (3%) (1)) Calculate the fellewing prebabilities: (5%) P(X a: 1), P(X=1), P(1 g X s“- 2), P(X :3 NZ), and P(1 e1“ X g ti) 7.(12%)In this preblem, we censider that three. calls are ebserved at a telephene switch, where veice calls (V) and data calls (D) are equally likely. Assume that X denetes the number cf veice calls, Y the number cf data calls. New let R = KY. (a) Find the cerrespending values cf the randem variables X, Y, and R. (3) (b) Find the prebability mass functicn (PMF) cf R. (3) (e) Find the expectatiun df R. (3) (d) Find the variance ef R. (3) 3.(Iﬂ%)ln this prcblem, we pmeeed with Problem 7. (a) Find the enpectatien ef the randem variable Q = RR) = 2R — 3x2. (2) (b) Find the secend mement cf the randem variable Q cf (a). (2) (c) Find the variance cf the randem variable Q cf (a). (2) (d) Find the standard deviatien cf the randem variable Q of (a). (2) (e) Which ene cf R and Q is mere cencentrated about their expectatiens and why '2' (2) 9.(10%)Suppese that, en the Richter scale, earthquakes cf magnitude 5.5 er higher have prebabilitv 0.015 ef damaging class—A bridges. Suppese that such intense earthquake eccur fellewing a Peissen tandem variable in average 1.5 times per year. If a class-A bridge is censtructed te last at least 60 years, what is the prebability that it will be undamaged by earthquakes far 60 years. ...
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