Ninety-nine percent of all babies survive delivery. However, 25 percent of all births involve Cesarean (C) sections, and when a Cesarean is performed, the baby survives 98 percent of the time. If a randomly chosen pregnant women does not have a C section, what is the probability that her baby survives? If a baby survives delivery, what is the probability that the mother had a C section?
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
1. Two dice are rolled. What is the
2. Suppose there are three urns. Urn 1 contains two red balls and three black balls. Urn 2 contains one red ball and three black balls. Urn 3 contains four red balls and two black balls. A ball is drawn at random from Urn 1. If it is red, a second ball is drawn at random from Urn 2, but if it is black, the second ball is drawn at random from Urn 3. a) What is the probability of the second ball being drawn from Urn 3? b) What is the probability that the second ball is black, given that it is drawn from Urn 2? c) What is the probability that the second ball is black? d) What is the probability that the second ball was drawn from Urn 3, given that it is black? e) What is the probability that the first ball drawn was red, given that the second ball drawn is red? f) What is the probability that both balls drawn are black?
3. A recent college graduate is planning to take the first three actuarial examinations in the coming summer. She will take the first actuarial exam in June. If she passes that exam, then she will take the second exam in July, and if she also passes that one, then she will take the third exam in September. If she fails an exam, then she is not allowed to take any others. The probability that she passes the first exam is 0.3. If she passes the first exam, then the conditional probability that she passes the second one is 0.5, and if she passes both the first and the second exams, then the conditional probability that she passes the third exam is 0.6. a) What is the probability that she does not pass all three exams? (Hint: Use the multiplication rule.) b) Given that she did not pass all three exams, what is the conditional probability that she passed the first and the second exams? c) Given that she did not pass all three exams, what is the conditional probability that she failed the first exam?
4. Urn 1 contains 2 white balls and 3 red balls, whereas Urn 2 contains 6 white and 2 red balls. Two balls are randomly chosen from Urn 1 and put into Urn 2, and a ball is then randomly selected from Urn 2. What is a) the probability that the ball selected from Urn 2 is white? b) the conditional probability that both transferred balls were white given that a white ball is selected from Urn 2?
5. 2 hands of 10 cards each are randomly selected from an ordinary deck of 52 playing cards. a) Compute the probability that each hand has exactly two hearts and four diamonds. b) Compute the probability that each hand has exactly two hearts, four diamonds and two clubs. c) Compute the probability that each hand has exactly two clubs given that each hand has exactly two hearts and four diamonds.
6. Ninety-nine percent of all babies survive delivery. However, 25 percent of all births involve Cesarean (C) sections, and when a Cesarean is performed, the baby survives 98 percent of the time. If a randomly chosen pregnant women does not have a C section, what is the probability that her baby survives? If a baby survives delivery, what is the probability that the mother had a C section?
7. ) A box has four drawers; the first drawer contains four gold coins, the second drawer contains one gold and three silver coins, the third drawer contains one gold and two silver coins and the fourth drawer contains two gold coins and one silver coin. Assume that one drawer is selected randomly and one of the two coins selected randomly from that drawer is gold and the other coin is silver. What is the probability that the chosen drawer is the one with one gold and three silver coins?
8. Suppose that each child born is equally likely to be a boy or a girl, independently of the sex distribution of the other children in the family. For a family having 7 children, compute the probabilities of the following events: a) Not all children are of the same sex. b) Three children are boys and four are girls. c) At most two are boys. d) There is at least one girl.
9. A simplified model of the movement of the price of a stock supposes that on each day the stock’s price either moves up 1 unit with probability 0.9 or moves down 1 unit with probability 0.1. The changes on different days are assumed to be independent. a) What is the probability that after four days the stock will be at its original price? b) What is the probability that after five days the stock’s price will have decreased by 1 unit? c) Given that the stock is at its original price after two days, what is the probability it will be below its original price after five days?
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