Sofia. Posted June 23, 2014 Report Share Posted June 23, 2014 Hi! I'm having some problem with the following exercise and I was hoping to get some expert-help 1. The probability that it rains in Raincity (wow, such imagination!) on any day of the year is 0.45. Calculate the probability that in any given week, it rains on exactly three consecutive days. Grateful for any help! Can't really call my teacher during the summer Link to post Share on other sites More sharing options...
ctrls Posted June 23, 2014 Report Share Posted June 23, 2014 (edited) You can model the number of days it rains in a week using a binomial distribution, as From there, gives the required probability. If you haven't done probability distributions yet, you could instead consider the probability of it raining on 3 days and not raining on 4 days, then multiply it by the number of ways this can occur. Edited June 23, 2014 by ctrls 1 Link to post Share on other sites More sharing options...
Rahul Posted June 23, 2014 Report Share Posted June 23, 2014 You can model the number of days it rains in a week using a binomial distribution, as From there, gives the required probability. If you haven't done probability distributions yet, you could instead consider the probability of it raining on 3 days and not raining on 4 days, then multiply it by the number of ways this can occur. I'm not sure about this, ctrls - it seems like that's the probability of it raining 3 days of the week, not 3 consecutive days! It seems intuitive to me that the correct result should be P(3 consecutive days | X=3)P(X=3)=(n(3 consecutive days)/n(3 days))P(X=3) where X~B(7,0.45). This takes into account the fact that not all options given by the binomial distribution are consecutive. If I can do my stars-and-bars right, n(3 consecutive days) seems to be 5!/4! and n(3 days) seems to be 7!/(3!4!). Hence, the required ratio is (5)/(7!/(3!4!))*P(X=3), which gives a result of P(X=3)/7, 0.0417, or 4.17%. 2 Link to post Share on other sites More sharing options...
Vioh Posted June 23, 2014 Report Share Posted June 23, 2014 Another way of explaining this without using binomial distribution (that gives the same result as that of Rahul) is that if it rains only 3 days in a week, then it does not rain in the other 4 days. Thus the probability is (0.45)3(1 - 0.45)4 = (0.45)3(0.55)4 = 0.00834 But there are only 5 ways of arranging 3 consecutive raining days, i.e. it can rain on:- Monday, Tuesday, Wednesday- Tuesday, Wednesday, Thursday- Wednesday, Thursday, Friday- Thursday, Friday, Saturday- Friday, Saturday, Sunday Thus the probability that it rains on exactly 3 consecutive days in any given week is: 0.00834 × 5 = 0.0417 = 4.17% Not so sure of this method though, so correct me if i'm wrong 1 Link to post Share on other sites More sharing options...
Sofia. Posted June 24, 2014 Author Report Share Posted June 24, 2014 (edited) Thanks very much! The answer you've all gotten (4,17%) is correct according to my textbook. EDIT: I though that I had got it, but when I tried the next one... well, I think I might need some help with that one too! 2. On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days? Edited June 24, 2014 by Sofia Karlsson Link to post Share on other sites More sharing options...
ctrls Posted June 24, 2014 Report Share Posted June 24, 2014 I'm not sure about this, ctrls - it seems like that's the probability of it raining 3 days of the week, not 3 consecutive days! It seems intuitive to me that the correct result should be P(3 consecutive days | X=3)P(X=3)=(n(3 consecutive days)/n(3 days))P(X=3) where X~B(7,0.45). This takes into account the fact that not all options given by the binomial distribution are consecutive. If I can do my stars-and-bars right, n(3 consecutive days) seems to be 5!/4! and n(3 days) seems to be 7!/(3!4!). Hence, the required ratio is (5)/(7!/(3!4!))*P(X=3), which gives a result of P(X=3)/7, 0.0417, or 4.17%. My fault for not reading the question properly, I completely missed the part about it raining only on consecutive days. Sorry about that. Link to post Share on other sites More sharing options...
Vioh Posted June 24, 2014 Report Share Posted June 24, 2014 2. On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days? Well since the event of him being able to watch the news is independent (i.e. whether Mr. Li is watches the news on day 1 or not doesn't really affect the probability on day 2). So we can use tree diagram here (it's quite easy to draw). The calculation is then very simple: 0.3 × 0.3 × 0.3 = 0.027 = 2.7% probability Hope this answer agrees with your textbook! Link to post Share on other sites More sharing options...
Sofia. Posted June 25, 2014 Author Report Share Posted June 25, 2014 2. On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days? Well since the event of him being able to watch the news is independent (i.e. whether Mr. Li is watches the news on day 1 or not doesn't really affect the probability on day 2). So we can use tree diagram here (it's quite easy to draw). The calculation is then very simple: 0.3 × 0.3 × 0.3 = 0.027 = 2.7% probability Hope this answer agrees with your textbook! It doesn't match: the textbook gives 0.05913. Link to post Share on other sites More sharing options...
Rahul Posted June 26, 2014 Report Share Posted June 26, 2014 2. On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days? Well since the event of him being able to watch the news is independent (i.e. whether Mr. Li is watches the news on day 1 or not doesn't really affect the probability on day 2). So we can use tree diagram here (it's quite easy to draw). The calculation is then very simple: 0.3 × 0.3 × 0.3 = 0.027 = 2.7% probability Hope this answer agrees with your textbook! It doesn't match: the textbook gives 0.05913. Unfortunately, I'm not getting that answer with any of the methods I can think of. Is there any more context to that question? I would think Vioh's answer is correct and it is miskeyed otherwise. Link to post Share on other sites More sharing options...
Sofia. Posted June 26, 2014 Author Report Share Posted June 26, 2014 2. On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days? Well since the event of him being able to watch the news is independent (i.e. whether Mr. Li is watches the news on day 1 or not doesn't really affect the probability on day 2). So we can use tree diagram here (it's quite easy to draw). The calculation is then very simple: 0.3 × 0.3 × 0.3 = 0.027 = 2.7% probability Hope this answer agrees with your textbook! It doesn't match: the textbook gives 0.05913. Unfortunately, I'm not getting that answer with any of the methods I can think of. Is there any more context to that question? I would think Vioh's answer is correct and it is miskeyed otherwise. I think the difference between this exercise and the first one is the lack of 'exactly'. The probability would be the sum of the probability to get home in time five days, the probability to get home on time four days where at least three days are consecutive (all but the Mon-Tues-Thurs-Fri) and the probability of only three consecutive days. That seems to give me 0.3^5 + (5!/4!1! - 1)*0.3^4*0.7 + 3*0.3^3*0.7^2, which was close to 0.0648 if I remember correctly. This feels right to me, but it still isn't the textbook answer. The textbook is probably misprinted (has happened way too often!), but which way is correct then? Link to post Share on other sites More sharing options...
Rahul Posted June 26, 2014 Report Share Posted June 26, 2014 (edited) If that is the way the question is worded - "On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days?" as opposed to "On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days in any given week?", then the answer Vioh gave of 2.7% is indeed correct. If it is worded in the second manner, however, the manner you are describing is correct as it takes into account the fact that it is indeed within one week. Hope that helps! Edited June 26, 2014 by Rahul Link to post Share on other sites More sharing options...
Sofia. Posted June 27, 2014 Author Report Share Posted June 27, 2014 If that is the way the question is worded - "On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days?" as opposed to "On a TV news channel, the evening news starts at the same time every day. The probability that Mr Li gets home from work in time to watch the news is 0.3. What is the probability that Mr Li gets home in time to watch the news on three consecutive days in any given week?", then the answer Vioh gave of 2.7% is indeed correct. If it is worded in the second manner, however, the manner you are describing is correct as it takes into account the fact that it is indeed within one week. Hope that helps! It does say in part a) that it is in a particular week, so I guess that goes for part b) as well. Thank you so much for all of your help! Link to post Share on other sites More sharing options...
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