tag:blogger.com,1999:blog-92159931930340527852015-08-16T07:52:21.039-07:00Monday ProbabilityCome every Monday, something will puzzle you. Come every Monday, something will challenge you. Come every Monday, something will sound interesting. Come every Monday, the beauty of the random world will engage you!
So, come every Monday.Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.comBlogger25125tag:blogger.com,1999:blog-9215993193034052785.post-57367489698515861982010-06-28T21:00:00.000-07:002010-06-28T21:00:51.740-07:00Monday Probability 25: PairsA row has a+b chairs. A group has 'a' girls and 'b' boys who sit on the chairs randomly. What is the expected number of pairs (a girl and a boy sitting next to each other)?Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-80000031978667934432010-06-21T18:49:00.000-07:002010-06-21T18:49:25.440-07:00Monday Probability 24: Around the CircleThere are m+1 nodes on a circle numbered 0 to m. You are initially at 0 and with equal probability can jump to one of your neighboring nodes. Find the probability that the i-th node is the last node visited, i.e. what is the probability that when node i is eventually visited for the first time, you have visited every other node at least once before that. Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-86304027664141637952010-06-13T21:20:00.000-07:002010-06-21T18:54:55.809-07:00Monday Probability 23: Three HeadsHow many flips on an average will produce three heads in a row?<br /><br />Answer: 14Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com1tag:blogger.com,1999:blog-9215993193034052785.post-3149039018284236512010-06-07T13:23:00.000-07:002010-06-07T13:23:54.703-07:00Monday Probability 22: CouponsThere are N distinct types of coupons in cereal boxes and each type is equally likely to be in any box. If a child collects n coupons, what is the expected number of distinct coupons types he has?Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-44487914330929830782010-05-31T22:08:00.000-07:002010-06-07T13:25:24.646-07:00Monday Probability 21: Boys and GirlsA company is holding a dinner for working mothers with at least one son. Ms. Jackson, a mother with two children, is invited. What is the probability that both children are boys?<br /><br />Answer: 1/3Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com3tag:blogger.com,1999:blog-9215993193034052785.post-65404999299334854932010-05-24T17:38:00.000-07:002010-05-24T17:38:44.577-07:00Monday Probability 20: Cars on RoadThe probability of observing at least one car on a highway during any 20-min time interval is 609/625. What is the probability of observing at least one car during any 5-min interval? Assume uniformity.Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-9497191887352574062010-05-16T23:28:00.000-07:002010-05-24T17:46:28.498-07:00Monday Probability 19: Increasing DiceWe throw 3 dice one by one. What is the probability that we obtain 3 points in strictly increasing order?<br /><br />Answer: 5/54Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-75204878466518708712010-05-11T11:57:00.000-07:002010-05-24T17:44:58.476-07:00Monday Probability 18: ChordsA point is chosen randomly inside a circle. A chord is drawn with the point as the midpoint. Find the probability that the length of the chord is less than the radius of the circle.<br /><br />Answer: 1/4Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-38389855863344135912010-05-04T10:42:00.000-07:002010-05-24T17:44:00.054-07:00Monday Probability 17: AdmitsIthaca college wants a class size of 150, ideally. Historically 30% of the admitted candidates have registered every year. This year the college sends out 450 admits. Find the probability of more than 150 people accepting the admits.<br /><br />Answer:Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-6445833671059933042010-04-26T13:20:00.000-07:002010-05-04T10:44:11.088-07:00Monday Probability 16: PostofficeYou enter a post-office and see two stations each serving a customer. The waiting time is exponentially distributed for both with mean 10 mins. You wait till one of the stations gets free. What is the probability of you leaving the postoffice last (after the two people being served)?<br /><br />Answer: 1/2Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-87584404353520698812010-04-19T13:15:00.000-07:002010-05-04T10:43:43.479-07:00Monday Probability 15: Making a TriangleWhat is the probability of making a triangle out of the three segments you get by breaking a thread of unit length at two points uniformly randomly?<br /><br />Answer: 0.25Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-19469793485769421332010-04-12T07:02:00.000-07:002010-05-24T17:40:45.957-07:00Monday Probability 14: Empty BoxesYou put k identical balls in n identical boxes, equally likely. What is the expected number of empty boxes?<br /><br />Answer: (n-1)^k / n^(k-1)Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-14428648290451832812010-04-05T07:44:00.000-07:002010-04-12T07:03:15.013-07:00Monday Probability 13: DartsMy probability of hitting a dart in the central circle is uniformly distributed between 0.4 to 0.6. Letâ€™s say I take the first shot and hit it inside the circle. What is the chance that my next hit goes inside the circle too?<br /><br />Answer: 38/75Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com2tag:blogger.com,1999:blog-9215993193034052785.post-81091942703163469062010-03-29T00:34:00.001-07:002010-04-05T10:40:45.780-07:00Monday Probability 12: Red before Blue and GreenAn urn has 10 red, 20 blue and 30 green balls. We start to withdraw balls without replacement. What is the probability of all reds coming out before atleast one blue and one green?<br /><br />Answer: 7/12Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-2231046925564318592010-03-22T09:26:00.000-07:002010-03-28T20:48:27.518-07:00Monday Probability 11: Random AreaThe length of the rectangle, x, is chosen uniformly from the interval [0,1]. Thereafter the width, y, is chosen uniformly between [0,x]. What is the expected area of the rectangle?<br /><br />Answer: 1/6Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-40221561457946249902010-03-15T07:16:00.000-07:002010-03-22T09:22:23.946-07:00Monday Probability 10: Loss of MemoryAt the maintenance shop one has to go to station A first and then to station B (in that order) before leaving. The waiting time at station A and B is exponentially distributed with means 6 and 3 minutes (i.e. on an average it takes 6 minutes to get served at station A and 3 minutes to get served at station B). When you arrive at the shop there is already somebody at station A but station B is empty. How long, on an average, will you take to get served at both the stations?<br /><br />Answer: 16 minutesMhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-36014305917029663682010-03-08T06:53:00.000-08:002010-03-15T07:06:59.763-07:00Monday Probability 9: Lizard in a TetrahedronA lizard is sitting at a vertex of a regular tetrahedron. At every move the lizard randomly selects one of the remaining vertices and moves to it. What is the probability that it will be back at the original vertex after n steps?<br /><br />Answer: <span class="TSrHSb"><span class="ze">[1 - (-3)^(1-n)] /4</span></span>Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-45620205646369487592010-03-01T00:32:00.000-08:002010-03-08T06:47:11.343-08:00Monday Probability 8: N GamblersIn a game of N gamblers, the i-th gambler starts the game with a_i dollars. In each round, two gamblers selected at random make a fair bet, and the winner gets a dollar from the loser. A gambler losing all his money leaves the game. The game continues as long as possible, i.e., until one of the gamblers has all the money. What is the expected number of rounds played?<br /><br />Answer: 1/2 * [(SUM(ai))^2 - SUM(ai^2)]Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-56755175902252415562010-02-22T00:57:00.000-08:002010-03-01T00:33:50.998-08:00Monday Probability 7: Balls and b bins<div style="text-align: justify;">There is an infinite supply of balls and b bins. Each ball is tossed and is equally likely to fall in any of the bins and the tosses are independent. Find the expected number of tosses before each bin has atleast one ball.<br /><br />Answer: b*(1+1/2+1/3+...+1/b) </div>Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com4tag:blogger.com,1999:blog-9215993193034052785.post-67546416719769550122010-02-15T06:13:00.000-08:002010-02-22T00:59:42.185-08:00Monday Probability 6: Candies in pocketYou are a candy lover and have n candies in your right pocket and m in your left. You chose one of the pockets randomly (equally likely) and eat one of the candies from that pocket. What is the porbability that you finish the candies in both pockets together, such that when you eat the last candy from one of the pockets the other pocket has atmost one more candy left?<br /><br />Answer: (m+n-2)C(m-1)/2^(m+n-3)Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com2tag:blogger.com,1999:blog-9215993193034052785.post-68327981118672136952010-02-08T12:58:00.000-08:002010-02-10T07:04:16.591-08:00Monday Probability 5: Variance of girls and boys<div style="text-align: justify;"><span style="font-size: small;"><span class="UIStory_Message">In a class with 18 boys and 12 girls, boys have probability 1/3 of knowing the answer and girls have probability 1/2 of knowing the answer to a typical question the teacher asks. Assuming that whether or not the students know the answer are independent events, find the <span class="text_exposed_hide"></span><span class="text_exposed_show">variance of the number of students who know the answer.</span></span></span><br /><br /><span style="font-size: small;"><span class="UIStory_Message"><span class="text_exposed_show">Answer: 7 </span></span></span></div>Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com2tag:blogger.com,1999:blog-9215993193034052785.post-91538453920930131362010-02-01T05:29:00.000-08:002010-02-22T01:00:28.392-08:00Monday Probability 4: Rolling dice thrice<div style="text-align: justify;">A dice is rolled and the number which comes can be taken as the prize money or you can ask for another roll. The new number is your prize money or you can ask for the final roll. In all cases the final roll is your prize money. What is the expected return for an optimal strategy?</div><br />Answer:14/3Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com2tag:blogger.com,1999:blog-9215993193034052785.post-8445810072707809722010-01-25T20:54:00.001-08:002010-01-25T20:55:46.309-08:00Monday Probability 3 : Drunkard on a PolygonA drunkard is on a n sided polygon, where n is even. He can go to either of the adjacent edges with equal probability. Find the expected number of steps he need to take to reach the opposite edge from where he started.<br /><br />Answer: n^2/4Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com2tag:blogger.com,1999:blog-9215993193034052785.post-23200103435603198612010-01-18T17:30:00.001-08:002010-06-21T18:57:36.107-07:00Monday Probability 2 : Birds on Wire<div style="text-align: left;"><span style="color: #333333; font-family: 'lucida grande',tahoma,verdana,arial,sans-serif;">On a black wire of unit length, birds come and sit at random, uniformly. You then take white paint and for each bird paint the wire between that bird and its nearest neighbor. What fraction of wire will be painted white after a while?</span><br /><span style="color: #333333; font-family: 'lucida grande',tahoma,verdana,arial,sans-serif;"><br /></span><br /><span style="color: #333333; font-family: 'lucida grande',tahoma,verdana,arial,sans-serif;">Answer: 7/18</span></div>Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com0tag:blogger.com,1999:blog-9215993193034052785.post-391785854816192922010-01-11T08:39:00.000-08:002010-01-25T20:56:20.161-08:00Monday Probability 1 : Noodles<h3 class="GenericStory_Message" data-ft="{"type":"msg"}"><span style="font-weight: normal;"><span style="font-size: medium;">A bowl of k number of noodles are cooked and are flexible. A trial consists of randomly choosing an end of a noodle for each hand, then gluing them together. Suppose you do k trials (same as number of noodles). What is the expected number of closed circuits?</span></span></h3><div>Answer: 1+1/3+1/5+...+1/(2k-1)<br /></div>Mhttp://www.blogger.com/profile/04296475349435748706noreply@blogger.com2