Probability (tutorial 3)

Tutorial 31.Find the probability that seven of 10 persons will recover from a tropical disease if we can assume independence and the probability is 0.8 that any one of them will recover from the disease.

0.2013

2.An automobile safety engineer claims that 1 in 10 automobile accidents is due to driver fatigue. What is the probability that at least 3 of 5 automobile accidents are due to driver fatigue?

0.0086

3.In a certain city, incompatibility is given as the legal reason in 70% of all divorce cases. Find the probability that five of the next six divorce cases will claim incompatibility as the reason.

0.3025

4. In a large population of aphids, 10% are type A and the remainder are type B. If five aphids are selected at random, what is the probability that they are all type B?

What is the smallest number of aphids that must be selected from the population if the probability that this sample contains at least one type A aphid is greater than 0.9? 0.5905; 22

5.A drawer contains a large number of socks of various colours, 50% of them are white. What is the least number of socks that have to be taken out of the box to ensure that the probability of taking at least 2 white socks is greater then 0.9? 7

6.The number of inquiries a person gets in response to a newspaper ad listing a piano for sale is a random variable having a Poisson distribution with mean 4.4. What are the probabilities that in response to such an ad a person will receivei) only two inquiries 0.1188ii) only three inquiries

0.1743iii) at most three inquiries?

0.351

7.The number of lost-and-found requests received each day by a railroad ticket office is a random variable having the Poisson distribution with mean 4.2. Find the probability that on any given day, the number of such requests will be

i)exactly five

0.1633

ii)at most seven

0.93618. There are two fire stations A and B. The number of calls at station A is a Poisson variable with mean of 3 calls a day and the number of calls at station B is a Poisson variable with mean of 4 calls a day. Find the probability that, on a certain day

i) exactly two calls are received by each station

0.0328

ii) exactly two calls are received in total by either or both stations 0.0223

9. In a large office building it is found that the number of light bulbs failing a day is a Poisson variable with a mean of two failures a day. Find the probability that

i)more than two light bulbs fail on any one day,

0.3233ii)more than five light bulbs fail during five consecutive days,

0.9329iii)more than two light bulbs fail on each day of a five-day week 0.0035

10a.When taping a television commercial, the probability that a certain actor will get his lines straight on any one take is 0.4. What is the probability that this actor will get his straight for the first time on the fourth take?

0.0864

b.The probability that a child exposed to a contagious disease will catch it is 0.7. Find the probability that the third child exposed to the disease will be the first to catch it.

0.0630c.The probability that a given person will believe a rumour is 0.25. Find the probability that the fifth person to hear the rumour will be the first one to believe it.

0.0791

11. According to medical research, the time between successive reports of a rare tropical disease is a random variable having the exponential distribution with mean 120 days. Find the probabilities that the time between successive reports of the disease will

i)exceed 240 days

0.1353ii)exceed 360 days

0.0498iii)be less than 60 days

0.393512. The mileage which car owners get with a certain kind of tyre is a random variable having an exponential distribution with mean 40000 miles. Find the probability that one of these tyres will last

i) at least 20000 miles

0.6065

ii) at most 30000 miles

0.527613. In a photographic process, the developing time of prints may be looked upon as a random variable having the normal distribution with mean 15.4 seconds and standard deviation 0.48 seconds. Find the probabilities that the time it takes to develop one of the prints will be

i) at least 16 seconds

0.1056

ii) at most 14.2 seconds

0.0062

iii) between 15 to 15.8 seconds.

0.5954

14.The weights and heights of group of children are each independently and normally distributed, with means of 35kg and 126cm and standard deviations of 10kg and 8cm. Find the probability that a child chosen at random will have

i) a weight more than 33kg

0.5793

ii) a height more than 122cm

0.6915

iii) both a weight more than 33kg and a height more than 122cm 0.4006

If nine children are picked at random, find the probability that six of them have a weight more than 33kg and a height more than 122cm.

0.0748

15.The times taken by individual members of a large group of people to complete a set task, are normally distributed with a mean of 10 minutes and a standard deviation of 2 minutes. Find the probability that three people chosen at random from this group will all complete the task in less than 8 minutes.

0.004

16.In a certain examination the marks are approximately normally distributed with a mean of 65 and a standard deviation of 10. Find the probability that an individual picked at random, has a mark from 60 to 69 inclusive.

0.3824

17.In a particular town, on average 51boys are born for every 49 girls born. By using the normal distribution as an approximation to the binomial distribution, find the probability that in 500 births there are less than 250 boys.

0.311418.A fair die is tossed 300 times. By using the normal distribution as an approximation to the binomial distribution, find the probability of getting more than 45 sixes.

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