Solutions to Exam 1, Ma441, Spr06 - Simmons 65 1 pts...

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Unformatted text preview: Simmons 65 1. pts.) Suppose, when looking at the hair color and eye color of an individual], that o 21 % have blond hair 0 37 % have brown eyes 0 2 % have blond hair and brown eyes In the problem below, consider randomly selecting an individual and recording their hair color and their eye color. In your computations below, please use: H ¥ H = flair is blond E E = Eyes are brown a. Are H, E disjoint? Why or why not? Hn E 4 ¢ (ma... Rave—blaml m: me (m) b. Show me, using the definition, that H, E are not independent (i.e. are dependent). ,oz: WHnE) 1: PM) WE) = (11.37): ,07770 W 50 [4,5 [194' RM c. What is the chance an individual has either blond hair or brown eyes? NHWE): rm2+V(s)»V(Ha._16) : .214— '37- (DZ. 7. .56' d. What is the chance an individu H = (Ff/‘5‘) u (we) PM): WH/lE‘) + Fame) L‘lme .21: PHI/159+— .01 \ '2 PHI/15:4): ,ZI-.o1=<.l7) e. If an individual has blond hair, what is the chance they have brown eyes? CEIH = V(E/)H)_ :25 : P ) PM) ‘ ~11 ’@ ‘ This is based on data collected by Snee, R. D. (1974), “Graphical display of two-way contingency tables”, i The American Statistician, 28, 9-12. I’ve rounded the numbers a bit to make the arithmetic easier. i l 2. (15 pts.) In the ‘Florida Lotto’ one selects 6 distinct numbers shown on a playing card showing the field of numbers 1, 2, . . ., 53. How much money you win depends on how many of your selected numbers match the 6 numbers eventually chosen as winners (these 6 numbers also being drawn without replacement). Here is the payout table for placing a $1 bet: FLORIDA LOTTO Number of Matches Estimate Prize (Gross Winnin_ ) $5 Write, in terms of binomial coefficients, the chance you have gross winnings of $70. {(541955 (amid/’42): 37°) -: €2< :7) (53 5 25 C 3. (96 pts) Suppose 5% of and 1% of are olor-blind Also suppose 2/3 of all - e, . s students at SDSM&T are m and 1/3 of tudents at DSM&T are female. i l Consider drawing an individual at random from the SDSM&T student population. a. What is the chance that the student is color-blind? / W5): FKCIMMMM flaw) You) : Cos)(%)* (“My 1:. b. If a color-blind student is selected, what‘is the chance that they are male? WM'C) : V(Clm)V(M) [7(a) 605)(1/3) (05M "6) +6“) C "3) X (26 pts.) Tetrahedral dice (i. e. dice in the shape of a pyramid), when thrown, are equally likely to show a 1, 2, 3, 4. a. A single tetrahedral die is tossed k times. What is the chance of at least one 2 appearing (your answer will be in terms of k)? b. A pair of tetrahedral dice are thrown k times. What is the chance of at least one double 2 appearing (your answer will be in terms of k)? (9 Wat/mm 1)= Mm 1) l’(~bcloubb. '2‘) (5" I5 1. (26pm) A former student of mine indicates the following game could be played at the “Clock Tower Lounge” in Rapid City: You bet 25¢ for the privilege of throwing 5 dice; only one play is allowed per day. Here is the associated payout table: “3 of a kind” means a particular number comes up 3 times and the other two dice take on other values. “4 of a kind” means a particular number comes up 4 times and the other dice is another value. . “3 of a kind” means all five dice take the same value. Problem: Determine the three probabilities, clearly labeling which is the chance for ‘3 of kind’, which is the chance for ‘4 of a kind’, and which is the chance for ‘5 of a kind’. Hint: When looking at, say, the ‘3 of a kind’ event, split it into ‘3 ones’ or ‘3 twos’ or. . . or ‘3 sixes’. ((30%?4 kid) ' V( 30% 93‘ 344495 a! moo- Effia) = H3 My)» V( 3414:»)4'H' “F W3 5‘“25) : (prawn (;)amr+~+ (WW .— éafitwm; ’ 4%)(éiéf a. I awe Withékml)‘: .455 .éH-E) mama W272}. m1): 6(§)(sz(§)°¥ p. 0007'? ...
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