\f\f\fTopic Assignment Grading Course: Name Topic Yours Name Professor's Name [optional] University 1 Topic Question 2. A president, vice president and a treasurer are to be chosen in a student club which has 40 members How many choices of the office are possible? How many choices if A will only serve if he is the president How many choices if A is not eligible to serve as president but can serve in other offices a. There are total 40 members for 3 positions so choices are possible 40P3= 40!/(40-3)! There would be only one president out of 59,280 ways b. There are again two cases: A serves as president, or F does not serve at all. In the first case, there are 1 P39,2 possibilities, while in the second there are P39,3. So the total number of possibilities is 1 P39,2 + P39,3 = 56,540 ways c. If A does not eligible at all for president but can serve in other offices as; 1*P39,2 = 1,482 ways Question 4. a. Each one can seat at all 8 eights of the row. So, total number of ways can seat if there is no restrictions as; Total number of ways: 8! i.e. 40,320 ways. 2 Topic b. If each couple sit next to each other then Its arranging 4 units and one for each couple. 4! = 24 Every couple can be seated in 2! ways but there are total 4 couples therefore, 2!*2!*2!*2!*4! = 384 Hence they can sit total 384 ways. Question 3. If we draw a band of 4 cards without replacement from the deck, what is the probability of getting a king or queen? Getting a queen out of 4 cards 1/4 Getting a king out of 4 cards 1/4 Draw 4 cards from the deck 4/10 Now, the probability of getting a king and a queen (1/4)*(1/4)/(4/10) = (5/32) Question 5. Given the digits 0,1,2,3,4,5,6. How many four digits number can be written? Note zero cannot take thousand positions for a four digit number. Any digit out of given 7 digits at thousand's place= 7 Non-zero number not at thousand place means there is 6 possible digits can take place at thousand place = 6 3 Topic Hundred's place = 6 (because '0' can also place and one digit already placed at thousand's place) Similarly for Tenth's place = 5 One's place = 4 So, possible four digits number = 6*6*5*4 => 720 positions Question 1. Suppose that 4 mobile phones are randomly chosen from a group of 5 new, 6 used but still working and 4 defective phones. If we let X and Y denote respectively, the number of new and used but still working phones that are chosen, find all the possible values of the random variable. Total phones = 5+6+4 => 15 Select 4 phones out of 15 = P15,4 All possible random values of used phones 15,408 ways and used 18,458 ways Question 7. Suppose X is a continuous random variable whose probability density function is given by What is the value of C? Please see attached screenshot for detail. Ans. c = 3/8 4 Topic Question 6 There are 23 = 8 possible outcomes. Let X be the number of heads observed. X is a random variable and can take the values 0,1,2 or 3. Since each outcomes is equally likely, the probability that 0 head is observed is P(X = 0) = ( 3 /0!) /8 = 1/8 . The probability that exactly 1 head is observed is P(X = 1) = ( 3/0 )/8 = 1/8 . The probability that exactly 2 heads are observed is P(X = 2) = ( 3/2 ) 8 = 3/8. The probability that exactly 3 heads are observed is P(X = 3) = ( 3/3 )/8 = 1/8. Table Random Value P( X= 0) P( X= 1) P( X= 2) P( X= 3) Probability 1/8 3/8 3/8 1/8 5 Topic Function P = (R/X)/N Where N= total possible outcomes X= Random variable R= head observation/tail observation. Graph Probability 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 P( X= 0) P( X= 1) P( X= 2) P( X= 3) 6