Set of elementary events (sample space) and formula of classical probability. Random or stochastic experiment; elementary random event; set of elementary events (sample space); random or stochastic event; impossible and certain (or sure) event

Topics of midterm 2020
Formulas
𝐢𝑛
π‘˜ =
𝑛!
π‘˜!βˆ™(π‘›βˆ’π‘˜)!
𝑃(𝐴|𝐡) =
𝑃(𝐴∩𝐡)
𝑃(𝐡)
𝑃(𝐡|𝐴) =
𝑃(𝐴∩𝐡)
𝑃(𝐴)
𝑃(𝐴) = 𝑃(𝐻1) βˆ™ 𝑃(𝐴|𝐻1) + 𝑃(𝐻2) βˆ™ 𝑃(𝐴|𝐻2) + β‹― + 𝑃(𝐻𝑛) βˆ™ 𝑃(𝐴|𝐻𝑛)
𝑃(𝐻𝑖|𝐴) =
𝑃(𝐻𝑖)βˆ™π‘ƒ(𝐴|𝐻𝑖)
𝑃(𝐴)
𝑃𝑛(π‘˜) = 𝐢𝑛
π‘˜ βˆ™ π‘π‘˜ βˆ™ (1 βˆ’ 𝑝)π‘˜ πΈπœ‰ = Ξ£ π‘₯𝑖 βˆ™ 𝑝𝑖
𝑛𝑖
=1 π·πœ‰ = Ξ£ (π‘₯𝑖 βˆ’ πΈπœ‰)2 βˆ™ 𝑝𝑖
𝑛𝑖
=1
1. Set of elementary events (sample space) and formula of classical probability.
Random or stochastic experiment; elementary random event; set of elementary events (sample space); random or stochastic event; impossible
and certain (or sure) event.
Probability; properties of probability; formula of classical probability.
Examples:
1. A coin is tossed three times. The result observed is the occurrence of heads (h) or tails (t) on the upward side of
the coin. Construct the set of elementary events. Calculate the probability of event
𝐴 = {𝑒π‘₯π‘Žπ‘π‘‘π‘™π‘¦ π‘œπ‘›π‘’ β„Žπ‘’π‘Žπ‘‘π‘  π‘‘π‘’π‘Ÿπ‘›π‘’π‘‘ 𝑒𝑝}.
2. Three cards are drawn at random from a pack 52 cards. Calculate the probability of event
𝐴 = {π‘Žπ‘‘ π‘™π‘’π‘Žπ‘ π‘‘ π‘‘π‘€π‘œ π‘Žπ‘π‘’π‘  𝑀𝑖𝑙𝑙 𝑏𝑒 π‘‘π‘Ÿπ‘Žπ‘€π‘›}.
3. You spin the spinner and flip a coin. Construct the set of elementary events. Use formula of classical
probability and calculate the probability of event
𝐴 = {π‘Žπ‘› π‘œπ‘‘π‘‘ π‘›π‘’π‘›π‘π‘’π‘Ÿ π‘Žπ‘π‘π‘’π‘Žπ‘Ÿπ‘’π‘‘ π‘Žπ‘›π‘‘ β„Žπ‘’π‘Žπ‘‘π‘  π‘‘π‘’π‘Ÿπ‘›π‘’π‘‘ 𝑒𝑝}.
4. Three balls are drawn at random from a box that has five white balls and two black balls. Calculate the
probability of event
𝐴 = {π‘Žπ‘‘ π‘™π‘’π‘Žπ‘ π‘‘ π‘‘π‘€π‘œ π‘€β„Žπ‘–π‘‘π‘’ π‘π‘Žπ‘™π‘™π‘  𝑀𝑖𝑙𝑙 𝑏𝑒 π‘‘π‘Ÿπ‘Žπ‘€π‘›}.
5. The examination program contains 50 questions. To pass the examination, every student must answer two of
them. One student has prepared the answers to 40 questions. What is the probability that he passed the
examination?
2. Theorems of probabilities.
Conditional probability; probability multiplication formulas of dependent and independent events; probability addition formulas of mutually
inclusive and mutually exclusive events; Bernoulli scheme.
Examples:
1. What is the probability of getting total of 6 or 12 when a pair of fair dice is tossed?
2. Twenty cards with numbers 1, 2, 3, …, 20 are placed on the table. One card is randomly drawn.
What is the probability of drawing a card with number multiple of 6 or even number?
3. In a city 40% of the families have a dog, 65% have a cat, and 10% have both a dog and a cat. A family is
chosen at random. What is the probability that this family has a dog or a cat?
4. Suppose two fair dice are tossed where each of the 36 possible outcomes is equally likely to occur.
Knowing that the first die shows a prime number, what is the probability that the sum of two dice equals 7?
5. Two riflemen are shooting at a target independently. The probability of hitting the target by the first rifleman is
equal to 0.9, by the second rifleman to 0.8. What is the probability that:
a) the target will be hit by one rifleman;
b) the target will be hit by two riflemen;
c) none of the riflemen will hit the target;
d) the target will be hit at least by one rifleman?
6. A system consists of three components as illustrated. If all the components function independently, and the
probability each component works is 0.9, what is the probability the entire system functions?
7. What is more likely to win two matches out of three or to win three matches out of five. Chance of winning one
match is 50%.
3. Law of total probability. Bayes’s formula.
Law of total probability; Bayes’ formula.
Examples:
1. Printer failures are associated with three types of problems: hardware, software, and electrical connections. A
printer manufacturer obtained the following probabilities from a database of tests results. The probabilities of
problems concerning hardware, software, and electrical connections are 0.1, 0.6, and 0.3, respectively. The
probability of a printer problem, given that there is a hardware problem is 0.9, given that there is a software
problem is 0.2, and given that there is an electrical problem is 0.5. If a customer seeks help to diagnose the
printer failure, what is the most likely cause of the problem?
2. Suppose that 5% of men and 0.25% of women are colour blind. Assume that there are an equal number of
males and females.
a) What is the probability that randomly selected person is colour blind?
b) A colour-blind person is chosen at random. What is the probability of this person’s being male?
4. Discrete random variable.
Discrete random variable: distribution law; probability distribution function and its properties. Numerical characteristics: expected value
(mean), mode, median, variance and standard variation.
Examples:
1. There are 5 balls in a box (one of them is white); 2 balls are selected at random. Let 𝑋 be the number of the
white balls taken. Write down the distribution law and the distribution function 𝐹(π‘₯). Sketch the graph of 𝐹(π‘₯).
Find expected value 𝐸𝑋, variance 𝐷𝑋,standard deviation πœŽπ‘‹, mode π‘€π‘œ probabilities
𝑃(𝑋 < 1),𝑃(𝑋 < 2), 𝑃(𝑋 β‰₯ 2) , 𝑃(0 ≀ 𝑋 < 1). Try two cases: a) sampling without replacement; b) sampling with replacement. 2. Two signals are transmitted independently. The probability of receiving the first signal correctly is equal to 0.8, and the probability of receiving the second signal correctly is equal to 0.9. Let 𝑋 be a number of correctly received signals. Write down the distribution law, find expected value 𝐸𝑋, variance 𝐷𝑋, mode π‘€π‘œ. 3. Let 𝑋 be a number of heads in 4 tosses of a fair coin. Write down the distribution law, find expected value 𝐸𝑋, variance 𝐷𝑋, mode π‘€π‘œ. What is the probability that heads appear less than two times? 5. Continuous random variable. Continuous random variable: probability distribution function and its properties, probability density function and its properties. Expected value (mean), mode, median, variance and standard variation. 1. Suppose that the error in the reaction temperature for a controlled laboratory experiment is a continuous random variable 𝑋 having the probability density function 𝑝(π‘₯) = { 0, π‘˜π‘Žπ‘– π‘₯ ≀ 0, 𝐢(1 βˆ’ π‘₯), π‘˜π‘Žπ‘– 0 < π‘₯ ≀ 1, 0, π‘˜π‘Žπ‘– π‘₯ > 1.
a) Find 𝐢;
b) Find 𝑃 (0 ≀ 𝑋 < 1 2 ); c) Find distribution function 𝐹(π‘₯); d) Calculate 𝐸𝑋, 𝐷𝑋, πœŽπ‘‹, π‘€π‘œ , 𝑀𝑒 . 2. Suppose a continuous random variable 𝑋 has the probability distribution function 𝐹(π‘₯) = { 1 8 0, π‘₯ ≀ 0, π‘₯3, 0 < π‘₯ ≀ 2, 1, π‘₯ > 2.
a) Find density function 𝑝(π‘₯).
b) Find 𝑃(1 < π‘₯ ≀ 3). 6. Discrete random two-dimensional vector. Distribution law of a two-dimensional discrete random vector in a tabular form; conditional distribution law, covariance, and correlation coefficient. Independence of two random variables. 1. A bag contains one black and three white balls. A sample of two balls is picked without replacement. Let 𝑋 and π‘Œ be the number of black and white balls, respectively, in the sample. a) Write down the distribution law (joint probability mass function) of 𝑋 and π‘Œ in tabular form. b) Find the probability that the sample contains at most one black and one white balls. c) Find the marginal distributions of the number of the white balls. d) Calculate 𝐸𝑋, πΈπ‘Œ, 𝐷𝑋, π·π‘Œ. e) Find the covariance between the number of white and black balls. f) Find the conditional probabilities, given that there are no white balls in the sample. 2. Two signals are transmitted independently. The probability to receive first signal correctly is equal to 0.8, and probability to receive the second signal correctly is equal to 0.9. Let 𝑋 and π‘Œ be the number of correctly received signals, respectively. Write down the distribution law (joint probability mass function) of 𝑋 and π‘Œ in tabular form. 3. The detail is non-defective if its length 𝑋 and its width π‘Œare within limits. The values of 𝑋 and π‘Œ are 𝑋 = { 0, 𝑖𝑓 π‘‘β„Žπ‘’ π‘™π‘’π‘›π‘”π‘‘β„Ž π‘œπ‘“ π‘‘β„Žπ‘’ π‘‘π‘’π‘‘π‘Žπ‘–π‘™ 𝑖𝑠 π‘€π‘–π‘‘β„Žπ‘–π‘› π‘‘β„Žπ‘’ π‘™π‘–π‘šπ‘–π‘‘π‘ , 1, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’. π‘Œ = { 0, 𝑖𝑓 π‘‘β„Žπ‘’ π‘€π‘–π‘‘π‘‘β„Ž π‘œπ‘“ π‘‘β„Žπ‘’ π‘‘π‘’π‘‘π‘Žπ‘–π‘™ 𝑖𝑠 π‘€π‘–π‘‘β„Žπ‘–π‘› π‘‘β„Žπ‘’ π‘™π‘–π‘šπ‘–π‘‘π‘ , 1, π‘œπ‘‘β„Žπ‘’π‘Ÿπ‘€π‘–π‘ π‘’. Write down the distribution law (joint probability mass function) of 𝑋 and π‘Œ in tabular form, if 5% of details are defective, 1% of them exceed the limits of length, 3% of them exceed the limits of width and 1% exceed the limits of length and width both. Order Now

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