According to The World Bank, only 9% of the population of Uganda had access to electricity as of 2009. Discrete Random Variables 4.1 Discrete Random Variables1 4.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize and understand discrete probability distribution functions, in general. Such a person wishes to buy a \(\$150,000\) one-year term life insurance policy. The amount of liquid in a \(12\)-ounce can of soft drink. Classify each random variable as either discrete or continuous. Let \(X\) denote the net gain from the purchase of a randomly selected ticket. \[0(0.969) + 5(0.025) + 25(0.005) + 100(0.001) = 0.35\]. The coin is a fair coin and is equally likely to land on heads or tails. \(X =\) the number of college and universities that offer online offerings. Basic. Ellen has music practice three days a week. Construct the probability distribution of \(X\). A multiple choice exam has \(20\) questions; there are four choices for each question. If one of the dice shows the number or object bet (and the other two do not show it), the player gets back his or her $1 bet, plus $1 profit. Suppose the “number” \(00\) is considered not to be even, but the number \(0\) is still even. Find the probability that her cats will wake her up no more than five times next week. Find the probability that at least one of them will be able to speak English. Each question has three possible choices for the answer. \(X\) is the number of hearts in a five-card hand drawn (without replacement) from a well-shuffled ordinary deck. If the card is a face card, you win $30. The chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. In one of its Spring catalogs, L.L. A class of \(130\) students meets in a classroom with \(130\) individual desks, exactly \(14\) of which are constructed for people who write with their left hands. every payday, at which time there are always two tellers on duty. What does it mean that the values zero, one, and two are not included for \(x\) in the PDF? We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Find the probability that more than five people in the sample are literate. He needs to ask someone directions. The average amount spent on electricity each July by a randomly selected household in a certain state. Recently, a nurse commented that when a patient calls the medical advice line claiming to have the flu, the chance that he or she truly has the flu (and not just a nasty cold) is only about 4%. How many residents do you expect will have adequate earthquake supplies? The tack is dropped and its landing position observed \(15\) times. One ticket will win \(\$2,000\), two tickets will win \(\$750\) each, and five tickets will win \(\$100\) each. On average, how many years do you expect it to take for an individual to earn a B.S.? A student takes a ten-question true-false quiz, but did not study and randomly guesses each answer. There are \((3)(4) = 12\) face cards and \(52 – 12 = 40\) cards that are not face cards. \(X =\) the number of children for a Spanish woman, Give the distribution of \(X\). Each page may be picked more than once. Use the following information to answer the next two exercises: The average number of times per week that Mrs. Plum’s cats wake her up at night because they want to play is ten. Find the probability that from \(8\) to \(12\) days will be lost next summer. We are interested in the number that offer distance learning courses. These are homework exercises to accompany the Textmap created for "Introductory Statistics" by Shafer and Zhang. \(X\) is the number of coins that match at least one other coin when four coins are tossed at once. Define the random variable \(X\) and give its distribution. If the player rolls doubles all three times there is a penalty. A work contains four errors. Solution. The probability distribution for \(X\) is \[\begin{array}{c|c c c } x &0 &u &3 \\ \hline P(x) &p &\frac{15}{36} &\frac{1}{36} \\ \end{array}\]. You buy a lottery ticket to a lottery that costs $10 per ticket. Describe the random variable \(X\) in words. Assuming that boys and girls are equally likely, construct the probability distribution of \(X\). What values does \(X\) take on? Legal. Find the probability that a person is audited more than twice. The prize is two passes to a Broadway show, worth a total of $150. \(−2\left(\dfrac{40}{52}\right)+30\left(\dfrac{12}{52}\right) = −1.54 + 6.92 = 5.38\). How do you know that? Find the probability that the student guesses more than 75% of the questions correctly. Let \(X =\) the number of people who have access to electricity. Find the probability that at most ten offer such courses. v~���d�������O�����X6H��������� ��jP���?�AKp��.������>}�b0�Ԡ��K�5����=����e�\�������U����#�g����9����k�j(? Why or why not? stream What are the values that \(X\) can take on? Define \(P(x)\), or the probability of \(x\). Over the long run of playing this game, what are your expected average winnings per game? Interpret the mean in the context of the problem. The pattern evident from parts (a) and (b) is that if. In a certain board game a player's turn begins with three rolls of a pair of dice. Eighty of the coupons are for a free gift worth $8. Find the expected value to the company of a single policy if a person in this risk group has a \(97.25\%\) chance of surviving one year. Of the next 25 patients calling in claiming to have the flu, we are interested in how many actually have the flu. Let \(X =\) the number of children married people have. Find the chance that he guesses correctly between four and seven times. According to the South Carolina Department of Mental Health web site, for every 200 U.S. women, the average number who suffer from anorexia is one. If the card is a face card, and the coin lands on Heads, you win $6, If the card is a face card, and the coin lands on Tails, you win $2. \(X =\) the number of adults in America who are surveyed until one says he or she will watch the Super Bowl. Let \(X =\) the number of people you test until you find a person infected with HIV. Explain. Let \(X =\) the number of people who are literate. These are homework exercises to accompany the Textmap created for "Introductory Statistics" by OpenStax. Explain what your calculations indicate about your long-term average profits and losses on this game. (a) Two random variables Xand Y are said to be correlated if and only if their covariance C XY is not equal to 0. Find the probability that she has fewer children than the Japanese average. Will the owner have the cover installed? The probability that all 25 not use the foil is almost zero. A “friend” offers you the following “deal.” For a $10 fee, you may pick an envelope from a box containing 100 seemingly identical envelopes. Compute the probability indicated. We are interested in the expected number of audits a person with that income has in a 20-year period. Discrete Random Variable . Find the probability that such a shipment will be accepted. What is the probability that the maternity ward will deliver more than five babies in one hour? I expect to break even. If the card is not a face card, you lose $2, no matter what the coin shows. \(P(x = 12) = 0.3186 P(x = 13) = 0.5882\) More likely to get 13. Find the average number of inferior quality grapefruit per box of a dozen. \(X =\) the number of audits in a 20-year period. The number of new cases of influenza in a particular county in a coming month. Investigate the relationship between independence and correlation. Each month a local blood bank sends an appeal to give blood to \(250\) randomly selected students. The average weight of newborn babies born in a particular county one month. A fair price for a ticket is $0.35. If not, explain why not. the number of times Mrs. Plum’s cats wake her up each hour.