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Given that three distinct integers are chosen at random from the first positive integers.
We need to compute the probability that (a) their sum is odd, (b) their product is even.
We know that sum of three integers is odd, if either all three integers are odd or one of the three integers is odd.
We know that the product of three integers is even if at least one of the three integers is even.
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- 12. The table shows the number of students who took the March Bar Test for the first time and thc numbcr of students who repact the exam. Passed Failed Total 6458 2058 First time 4427 2252 1845 407 Repeat Total 8737 4834 3903 a. )Find the probability that a student failed, given that the student took the exam for the first time. b.) Find the probability that a studnt repeated the exam, given the student passed.Find a general solution to the given Cauchy-Euler equation for t> 0. d²w 8 dw dt² t dt 6 W=0 The general solution is w(t) = 0B positive: 24 B negative: 6 hat is the probability that a rand 0 Has Type O blood? ) Is Rh-negative? Has Type O blood and is Rh-ne | Has Type O blood or is Rh-neg Has Type O blood, given that t Is Rh-negative, given that they
- VSC Type here to search F1 W TE R Boys Not boys Juniors 65 35 Not juniors 220 145 Total 285 180 465 1. What is the probability that a student selected at random is a boy? PB + )= 285/465 2. What is the probability that a boy selected at random is a Junior? P(| J|B)= 65/285 3. What is the probability of randomly choosing a student who is a Junior and a boy? P(JB) WOMENGEN 65/465 4. What is the probability of randomly choosing a student who is a girl given that she is not junior? P(| G|J = 145/365 5. What is the probability of randomly selecting a girl? P(G) = 180/965 31 ( 99+ F5 1. I T Y F2 F3 1² 1 Total % 100 365 My Apps Dashboar... 220 F6 F7 F8 1'. 1'. 65 F9 35 Cip F10 ETE 145 F11 F12 4- Backspaceof Suppose a child is born to two parents who are cystic fibrosis carriers. Construct a chart of gene combinations to find the probability that the child will neither have nor carry the disease. P(neither have nor carry the disease) = (Type an integer or a simplified fraction.) of of y of of C Time Remaining: 01-0637Computers A and B can independently generate positive integers: • Computer A generates positive integer n with probability a/3", where a is a constant. • Computer B generates positive integer n with probability 3/n!, where ß is a constant. (a) Find the probability that the number generated by A is larger than the number generated by B. (b) Find the probability that the number generated by A is smaller than the number generated by B.
- K Suppose that there is a white um containing four white balls and two red balls and there is a red urn containing two white balls and three red balls. An experiment consists of selecting at random a ball from the white urn and then (without replacing the first ball) selecting at random a ball from the urn having the color of the first ball. Find the probability that the second ball is red. For this problem, assume that OA. 5 represents drawing a white ball from an um and that R represents drawing a red ball from an urn. Draw a tree diagram that represents this situation. Choose the correct answer below. OD So, the probability of the second ball being red is (Type an integer or a simplified fraction.) OB. X OC. -RA drawer holds purple socks and yellow socks. If n socks are taken out of the drawer at random, the probability that all are yellow is 1/2. What is the smallest possible number of socks in the drawer (as a function of n)?Q3 Suppose we have two boxes and 2N cards, of which N are blue and N are red. Initially, N of the cards are placed in box 1, and the remainder of the cards are placed in box 2. At each trial a card is chosen at random from each of the boxes, and the two cards are put back in the opposite boxes. Let Xo denote the number of blue cards initially in box 1 and, let Xn for n ≥ 1, be the number of blue cards in box 1 after the nth trial. Find the closed form transition probailities of the Markov chain Xn, n ≥ 0.
- Consider 10 independent tosses of a biased coin with the probability of Heads at each toss equal to p, where 0 < p < 1. (a) Let A be the event that there are exactly 6 Heads in the first 8 tosses. Let B be the event that the 9th toss results in Heads. Find P(B | A) and express it in terms of p. (b) Find the probability that there are exactly 3 Heads in the first 4 tosses and exactly 2 Heads in the last 3 tosses. Express your answer in terms of p. (c) Given that there were exactly 4 Heads in the first 7 tosses, find the probability that the 2nd Heads occurred at the 4th toss. Give a numerical answer. (d) Find the probability that there are exactly 5 Heads in the first 8 tosses and exactly 3 Heads in the last 5 tosses.2:51 A flush in the card game of poker occurs if a player gets five cards that are all the same suit (clubs, diamonds, hearts, or spades). Complete parts (a) and (b) to obtain the probability of being dealt a flush in five cards. (a) Initially concentrate on one suit, say clubs. There are 13 clubs in a deck. Compute P(five clubs) = P(first card is clubs and second card is clubs and third card is clubs and fourth card is clubs and fifth card is clubs). P(five clubs) = 0.000495 (Round to six decimal places as needed.) (b) A flush can occur if a player receives five clubs or five diamonds or five hearts or five spades. Compute P(five clubs or five diamonds or five hearts or five spades). Note that the events are mutually exclusive. P(five clubs or five diamonds or five hearts or five spades) = (Round to three decimal places as needed.) Vo) 1 LTE 4G 1% ||| = OA box in a supply room contains 24 compact fluorescent lightbulbs, of which 8 are rated 13-watt, 9 are rated 18-watt, and 7 are rated 23-watt. Suppose that three of these bulbs are randomly selected. (Round your answers to three decimal places.) *****-+ in the probability it exactly two of the selected hulhe 1 na is the ph that all the -ina? "'? (d) If bulbs are selected one by one until a 23-watt bulb is obtained, what is the probability that it is necessary to examine at least 6 bulbs?