Unit06:Probability
IF Probability
151,23,415,6
i5
PIE 1,9 f
ii e45,6
PE
iii 2,4,6
PLE
2s1,1 1,2 1,3 0,6
151 36
i1,4 4,1 2,3 3,2
PIE 15
ii 6,2 2,6 5,3 3,5 4,4
PIE 3
3151 52
ip1414632
44 42720985
ii El 242 36
PLE 248725
iii El Cy 715
PCE 2725 465
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Probability and Distributions: Exercises and Problems, Study notes of Business Mathematics
A collection of exercises and problems related to probability and distributions. It covers various concepts, including probability calculations, finding probabilities of events, and applying probability concepts to real-world scenarios. Suitable for students studying probability and statistics, providing them with practice problems to enhance their understanding of the subject.
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Probability and Distributions
A uniform die is thrown at random. Find the probability that the number on it is :
(i) 5, (ii) greater than 4, (iii) even.
- In a single throw with two uniform dice find the probability of throwing(i) Five, (ii)
Eight.
- Four cards are drawn at random from a pack of 52 cards. Find the probability that
(i) They are a king, a queen, a jack and an ace.
(ii) Two are kings and two are aces.
(iii) All are diamonds.
(iv)Two are red and two are black.
(v) There is one card of each suit.
(vi)There are two cards of clubs and two cards of diamonds.
- What is the chance that a non-leap year should have fifty-three sundays?
- A bag contains 20 tickets marked with numbers 1 to 20. One ticket is drawn at random.Find
the probability that it will be a multiple of (i) 2 or 5, (ii) 3 or 5.
- An urn contains 8 white and 3 red balls. If two balls are drawn at random, find the
probability that
(i) both are white, (ii) both are red, (iii) one is of each colour.
- Tickets are numbered from 1 to 100. They are well shuffled and a ticket is drawn at
random. What is the probability that the drawn ticket has :
(a) an even number? (b) a number 5 or a multiple of 5? (c) a number which is greater
than 75? (d) a number which is a square?
- The Federal Match Company has forty female employees and sixty male employees. If two
employees are selected at random, what is the probability that both will be males,
(ii) both will be females, (iii) there will be one of each sex?
- A committee of 4 persons is to be appointed from 3 officers of the production department,
4 officers of the purchase department, two officers of the sales department and 1 chartered
accountant. Find the probability of forming the committee in the following manner :
(i) There must be one from each category
(ii) It should have at least one from the purchase department
(iii)The chartered accountant must be in the committee
- A committee of four has to be formed from among 3 economists, 4 engineers, 2
statisticians and 1 doctor. (i) What is the probability that each of the four professions is
represented on the committee? (ii) What is the probability that the committee consists
of the doctor and at least one economist?
- A Chartered Accountant applies for a job in two firms X and Y. He estimates that the
probability of his being selected in firm X is 0·7, and being rejected at Y is 0·5 and the
probability of at least one of his applications being rejected is 0·6. What is the probability
that he will be selected in one of the two firms?
- Probability that a man will be alive 25 years hence is 0·3 and the probability that his wife
will be alive 25 years hence is 0·4. Find the probability that 25 years hence (i) both will
0·17 and 0·23 and the probability that its value will decrease by Re. 0·25 per share is 0·14,
what is the expected gain per share?
- The hourly wages of 1,000 workmen are normally distributed around a mean of Rs. 70 and
with a standard deviation of Rs. 5. Estimate the number of workers whose houly wages
will be: (i) between Rs. 69 and Rs. 72 (ii) more than Rs. 75 ; (iii) less than Rs. 63.
(iv) Also estimate the lowest hourly wages of the 100 highest paid workers.
- The average daily sale of 500 branch offices was Rs. 150 thousand and the standard
deviation Rs. 15 thousand. Assuming the distribution to be normal indicate how many
branches have sales between :
(i) Rs. 120 thousand and Rs. 145 thousand? (ii) Rs. 140 thousand and Rs. 165
thousand?
- As a result of tests on 2,000 electric bulbs manufactured by a company, it was found that
the lifetime of the bulb was normally distributed with an average life of 2,040 hours and
standard deviation of 60 hours. On the basis of the information, estimate the number of the
bulbs that is expected to burn for
(a) more than 2,150 hours, and (b) less than 1,960 hours.
- A project yields an average cash flow of Rs. 500 lakhs with a standard deviation of Rs.
80 lakhs. Calculate the following probabilities assuming the normal distribution :(i)
Cash flow will be more than Rs. 550 lakhs (ii) Cash flow will be less than Rs. 440 lakhs
(iii) Cash flow will be between Rs. 450 lakhs and Rs. 530 lakhs.
- The height of students of a college is found to have normal distribution with mean 162.
cm and s.d 6 cm. Find the probability that a student selected at random will have
i. height greater than 168 cm
ii. height less than or equal to 150 cm
iii. height between 150 cm and 168 cm
- A sales tax officer has reported that the average sales of the 500 business that he has todeal
with during a year amounted to Rs.36000 with a SD of Rs.10000. Assuming that the sales
in these business are normally distribution,
i. The number of business the sales of which are over Rs.
ii. The percentage of business the sales of which are likely to range between
Rs.30000 & Rs.
- As a result of tests on 20000 electric bulbs manufactured by a company; it was found that
the life tone of the bulb was normally distributed with an average life of 2040 hours and
standard deviation of 60 hours on the basis of the information estimate the number of
bulbs that are expected to burn for,
i. More than 2150 hrs
ii. Less than 1960 hrs
iii. Between 1687hrs& 1986hrs.
- A bank manager has learned that the length of time the customer has to wait for being
attended by the letter is normally distributed with mean time of 5 minutes and standard
deviation of 0.8 minutes. Find the probability that the customer has to wait
i. less than 6 minutes
ii. more than 3.5 minutes
iii. b/w 3.4 and 6.2 minutes
- 15000 students appeared for an examination the mean marks were 49 and standard
deviation of marks was 6. Assuming the marks to be normally distributed what
proportion of students scored
i. more than 55 marks
ii. less than 45 marks
- A production engineer finds that an average mechanics working in a machine shop
complete a certain task in 15 minutes. The time required to complete the task is
approximately normally distributed with standard deviation of 3 minutes find the
probability that the task is completed
i. less than 8 minutes.
ii. more than 9 minutes.
iii. B/W 10 & 12 minutes.
- The average selling prices of cell phone in a shop is `. 25,000 with standard deviation is
3,000 assuming the distribution of cell phone price to be normal distribution. What is the
probability that the cell phone price
i. more than `26,
ii. less than `18,
iii. between 20,000 and28,
An aptitude test for selecting officers in an insurance company was conducted to 1000
candidates the average score was found to be 42 with a standard deviation of 24.
Assuming normal distribution of scores find how many would have scored
i. more than 60.
ii. B/W 36 & 60.
iii. less than 35.
- A soft drink machine is regulated so that it discharges an average of 200 ml per cup ifthe
quantity of drink is normally distributed with a s.d equal to 15 ml
i. what proportion of the cups will contain quantity more than 224 ml
ii. what is probability that a cup contains between 191 ml & 209 ml
iii. what is the chance that machine fills less than 215 ml.
- Suppose that during periods of transcendental meditation the reduction of a person’s
oxygen consumption is a random variable having a normal distribution with mean 37.
cc per minute and S.D. 4.6 cc per minute. Find the probability that during a period of
transcendental meditation a person’s oxygen consumption will reduced by
i. at most 44.5 cc per minute
ii. at least 35.0 cc per minute
iii. anywhere from 30.0 to 40.0 cc per minute.