Basic probability (advanced practice)

{"questions":[{"content":"In the collection of question cards for biology there are $45$ question cards, in $9$ of them there is a question on the topic “Botany”. Find the probability that a student will get a question on the topic “Botany” on a randomly selected exam question card.[[fill_choice_big-1]][[image-16]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$0.5$","$0.8$","$0.2$"],"placeholder":0,"answer":2},"image-16":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/09/mineralnye-udobreniya-kolba-banka-stelo.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","$$P(A)=\\frac{9}{45}=0.2$$"]},{"content":"On average, $22$ out of every $1\\space000$ garden pumps offered for sale leak. Find the probability that one randomly selected garden pump does not leak.[[fill_choice_big-2]][[image-51]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$0.978$","$0.022$","$0.088$"],"placeholder":0,"answer":0},"image-51":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/podsolnuh-01-1.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of pumps that do not leak: $$1\\space000-22=978$$","$$P(A)=\\frac{978}{1\\space000}=0.978$$"]},{"content":"The taxi company has $50$ passenger cars in stock: $34$ of which are white with yellow inscriptions on the sides, the rest are yellow with white inscriptions. Find the probability that a yellow car with white inscriptions will respond to a random call.[[fill_choice_big-3]][[image-101]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$0.32$","$0.39$","$0.45$"],"placeholder":0,"answer":0},"image-101":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/10/gorod-lyudi-zanyatoj-mashiny-1-1.svg","width":"351"}},"step":1,"hints":["Find the number of cars with white inscriptions: $$50-34=16$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","$$P(A)=\\frac{16}{50}=0.32$$"]},{"content":"$25$ athletes are participating in the gymnastics championship: $6$ from Russia, $7$ from Australia, the rest from Germany. The order in which the gymnasts perform is determined by lot. Find the probability that the athlete competing first is from Germany.[[fill_choice_big-4]][[image-180]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$0.35$","$0.26$","$0.48$"],"placeholder":0,"answer":2},"image-180":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/10/run.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of athletes from Germany: $$25-6-7=12$$","$$P(A)=\\frac{12}{25}=0.48$$"]},{"content":"Scientific conference is held in $5$ days. A total of $75$ talks are planned - the first three days of $13$ talks, the rest are distributed equally between the fourth and fifth days. A talk by Professor $M.$ is planned at the conference. The order of talks is determined by drawing lots. What is the probability that Professor $M.$'s talk will be scheduled for the last day of the conference?[[fill_choice_big-5]][[image-261]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$0.24$","$0.35$","$0.42$"],"placeholder":0,"answer":0},"image-261":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/12/pisatel-01.svg","width":"348"}},"step":1,"hints":["In the first three days there are $13$ talks, which means in the last two days: $$75-3\\cdot13=36$$","In the last two days, talks are divided equally: $$36:2=18$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes: $$P(A)=\\frac{18}{75}=0.24$$<br />"]},{"content":"There is a person in the group of tourists $28$. They are transported to a remote area by boat, carrying $7$ people per trip. The order in which the boat carries tourists is random. Find the probability that tourist $P.$ will arrive on the last boat trip.[[fill_choice_big-6]][[image-298]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$0.375$","$0.3$","$0.25$"],"placeholder":0,"answer":2},"image-298":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/07/turist-palatka-kemping-3.svg","width":"350"}},"step":1,"hints":["In total the boat will make $4$ voyages, the probability of getting on one of the $4$ voyages is: $1$ to $4.$","$$P(A)=\\frac{1}{4}=0.25$$"]},{"content":"One number is chosen at random from the set of positive integers from $25$ to $34$. What is the probability that it is divisible by $4?$[[fill_choice_big-7]][[image-339]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$0.3$","$0.4$","$0.2$"],"placeholder":0,"answer":2},"image-339":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/04/uchitel-doska-globus.svg","width":"350"}},"step":1,"hints":["In total, between $25$ and $34$ there are $10 $ numbers. Among them, two numbers are divisible by $4$:$$28,32$$","$$P(A)=\\frac{2}{10}=0.2$$"]},{"content":"In the collection of question cards for mathematics there are $35$ question cards, in $14$ of them there is a question on the topic “Rational numbers”. Find the probability that a student will not get a question on the topic “Rational Numbers” on a randomly selected exam question card.[[fill_choice_big-8]][[image-384]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$0.4$","$0.6$","$0.75$"],"placeholder":0,"answer":1},"image-384":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/10/kniga_nauka-chitaet-obrazavr-01-01.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of question cards that do not contain a question on the topic “Rational Numbers”: $$35-14=21$$","$$P(A)=\\frac{21}{35}=0.6$$"]},{"content":"The factory produces bags. On average, $88$ bags from $100$ do not have hidden defects. Find the probability that the purchased bag will be defective.[[fill_choice_big-9]][[image-433]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$0.12$","$0.88$","$0.08$"],"placeholder":0,"answer":0},"image-433":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/05/kniga_obrazavr-01.svg","width":"348"}},"hints":["Probability that the bag will be defective: $$100-88=12$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes: $$P(A)=\\frac{12}{100}=0.12$$"]},{"content":"In a random experiment, a fair coin is tossed twice. Find the probability that tails will appear exactly once.[[fill_choice_big-10]][[image-486]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$0.2$","$0.5$","$0.25$"],"placeholder":0,"answer":1},"image-486":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/11/dengi.svg","width":"350"}},"step":1,"hints":["List the possible events: $$HH, HT, TH, TT$$","Tails appears exactly once in two of the four outcomes:$$P(A)=\\frac{2}{4}=0.5$$"]}]}

{"questions":[{"content":"In the collection of question cards on biology there are $35$ question cards, in $28$ of them there is a question on the topic “Zoology”. Find the probability that a student will get a question on the topic “Zoology” on a randomly selected exam question card.[[fill_choice_big-1]][[image-16]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$0.5$","$0.8$","$0.2$"],"placeholder":0,"answer":1},"image-16":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/09/dnk-uchenyi-01.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","$$P(A)=\\frac{28}{35}=0.8$$"]},{"content":"On average, $32$ out of every $1\\space000$ garden hoses offered for sale leak. Find the probability that one randomly selected garden hose does not leak.[[fill_choice_big-2]][[image-51]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$0.968$","$0.032$","$68$"],"placeholder":0,"answer":0},"image-51":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/09/polivaet-rasteniya-ogorod-rastenie-1-1.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of hoses that do not leak: $$1\\space000-32=968$$","$$P(A)=\\frac{968}{1\\space000}=0.968$$"]},{"content":"The taxi company has $50$ passenger cars in stock: $29$ of which are black with yellow inscriptions on the sides, the rest are yellow with black inscriptions. Find the probability that a yellow car with black inscriptions will respond to a random call.[[fill_choice_big-3]][[image-101]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$0.42$","$0.29$","$0.55$"],"placeholder":0,"answer":0},"image-101":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/taksi.svg","width":"351"}},"step":1,"hints":["Find the number of cars with black inscriptions: $$50-29=21$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","$$P(A)=\\frac{21}{50}=0.42$$"]},{"content":"$28$ athletes are participating in the gymnastics championship: $11$ from Russia, $10$ from the USA, the rest from China. The order in which the gymnasts perform is determined by lot. Find the probability that the athlete competing first is from China.[[fill_choice_big-4]][[image-180]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$0.05$","$0.36$","$0.25$"],"placeholder":0,"answer":2},"image-180":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/12/fehtovanie-01.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of athletes from China: $$28-11-10=7$$","$$P(A)=\\frac{7}{28}=0.25$$"]},{"content":"Scientific conference is held in $5$ days. A total of $75$ talks are planned - the first three days of the $23$ talk, the rest are distributed equally between the fourth and fifth days. A talk by Professor $M.$ is planned at the conference. The order of talks is determined by drawing lots. What is the probability that Professor $M.$'s talk will be scheduled for the last day of the conference?[[fill_choice_big-5]][[image-261]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$0.04$","$0.05$","$0.32$"],"placeholder":0,"answer":0},"image-261":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/konferentsiya-01-3.svg","width":"348"}},"step":1,"hints":["In the first three days the talk is carried out according to $23$, which means in the last two days: $$75-3\\cdot23=6$$","In the last two days, talks are divided equally: $$6:2=3$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes: $$P(A)=\\frac{3}{75}=0.04$$<br />"]},{"content":"There is a person in the group of tourists $32$. They are delivered to a remote area by helicopter, carrying $8$ people per flight. The order in which the helicopter transports tourists is random. Find the probability that tourist $P.$ will take the last helicopter flight.[[fill_choice_big-6]][[image-298]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$0.375$","$0.3$","$0.25$"],"placeholder":0,"answer":2},"image-298":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/03/vertolet-1.svg","width":"350"}},"step":1,"hints":["In total, the helicopter will make $4$ departure, the probability of being on one of the $4$ departures: $1$ to $4.$","$$P(A)=\\frac{1}{4}=0.25$$"]},{"content":"One number is chosen at random from the set of positive integers from $20$ to $29$. What is the probability that it is divisible by $4?$[[fill_choice_big-7]][[image-339]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$0.3$","$0.4$","$0.2$"],"placeholder":0,"answer":0},"image-339":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/08/doska-magnitnaya.svg","width":"350"}},"step":1,"hints":["In total, between $20$ and $29$ there are $10 $ numbers. Among them, three numbers are divisible by $4$:$$20,24,28.$$","$$P(A)=\\frac{3}{10}=0.3$$"]},{"content":"In the collection of mathematics question cards there are $15$ question cards, in $9$ of them there is a question on the topic “Inequalities”. Find the probability that a student will not get a question on the topic “Inequalities” on a randomly selected exam question card.[[fill_choice_big-8]][[image-384]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$0.4$","$0.6$","$0.45$"],"placeholder":0,"answer":0},"image-384":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/08/pishet-formuly-na-doske-doska.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of question cards that do not contain a question on the topic “Inequalities”: $$15-9=6$$","$$P(A)=\\frac{6}{15}=0.4$$"]},{"content":"The factory produces shoes. On average, $85$ pairs of shoes made from $100$ do not have hidden defects. Find the probability that the purchased shoes will be defective.[[fill_choice_big-9]][[image-433]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$0.15$","$0.95$","$0.05$"],"placeholder":0,"answer":0},"image-433":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/05/ryukzak.svg","width":"348"}},"hints":["The probability that the shoes will be defective: $$100-85=15$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes: $$P(A)=\\frac{15}{100}=0.15$$"]},{"content":"In a random experiment, a fair coin is tossed twice. Find the probability that you will never get tails.[[fill_choice_big-10]][[image-486]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$0.2$","$0.5$","$0.25$"],"placeholder":0,"answer":2},"image-486":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/11/svobodnoe13.svg","width":"350"}},"step":1,"hints":["List the possible events: $$HH, HT, TH, TT$$","There are no tails in one of the four outcomes:$$P(A)=\\frac{1}{4}=0.25$$"]}]}

{"questions":[{"content":"In the collection of question cards for biology there are $55$ question cards, in $11$ of them there is a question on the topic “Anatomy”. Find the probability that a student will get a question on the topic “Anatomy” on a randomly selected exam question card.[[fill_choice_big-1]][[image-16]]","widgets":{"fill_choice_big-1":{"type":"fill_choice_big","options":["$5$","$0.8$","$0.2$"],"placeholder":0,"answer":2},"image-16":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/11/uchenyi-kletochnaya-membrana-mikroskop-01-01.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","$$P(A)=\\frac{11}{55}=0.2$$"]},{"content":"On average, $5$ out of every $1\\space000$ garden sprinklers offered for sale leak. Find the probability that one randomly selected garden sprinkler does not leak.[[fill_choice_big-2]][[image-51]]","widgets":{"fill_choice_big-2":{"type":"fill_choice_big","options":["$0.995$","$0.005$","$200$"],"placeholder":0,"answer":0},"image-51":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/08/polivaet-rastenie-1.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of non-leaking sprinklers: $$1\\space000-5=995$$","$$P(A)=\\frac{995}{1\\space000}=0.995$$"]},{"content":"The probability that a new ballpoint pen writes poorly or does not write at all is equal to $0.21.$ The buyer, without looking, takes one ballpoint pen from the box. Find the probability that this pen writes well.[[fill_choice_big-3]][[image-101]]","widgets":{"fill_choice_big-3":{"type":"fill_choice_big","options":["$0.79$","$0.21$","$0.5$"],"placeholder":0,"answer":0},"image-101":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/05/ruchka.svg","width":"351"}},"step":1,"hints":["Find the probability of the complementary event.","$$1-0.21=0.79$$"]},{"content":"The lifting competition includes $4$ athletes from Finland, $7$ athletes from Denmark, $9$ athletes from Sweden and $5$ from Norway. The order in which the athletes compete is determined by lot. Find the probability that the athlete competing last is from Sweden.[[fill_choice_big-4]][[image-180]]","widgets":{"fill_choice_big-4":{"type":"fill_choice_big","options":["$0.05$","$0.16$","$0.36$"],"placeholder":0,"answer":2},"image-180":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/08/sportsmeni-shtanga-2.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Total athletes in the competition: $$4+7+9+5=25$$","$$P(A)=\\frac{9}{25}=0.36$$"]},{"content":"Scientific conference is held in $5$ days. A total of $75$ talks are planned - the first three days of $17$ talks, the rest are distributed equally between the fourth and fifth days. A talk by Professor $F.$ is planned at the conference. The order of talks is determined by drawing lots. What is the probability that Professor $F.$'s talk will be scheduled for the last day of the conference?[[fill_choice_big-5]][[image-261]]","widgets":{"fill_choice_big-5":{"type":"fill_choice_big","options":["$0.16$","$0.25$","$0.42$"],"placeholder":0,"answer":0},"image-261":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2026/01/teacher.svg","width":"348"}},"step":1,"hints":["In the first three days there are $17$ talks, which means in the last two days: $$75-3\\cdot17=24$$","In the last two days, talks are divided equally: $$24:2=12$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes: $$P(A)=\\frac{12}{75}=0.16$$<br />"]},{"content":"Before the start of a football match, the referee tosses a coin to determine which team will start with the ball. The Khimik team plays three matches with different teams. Find the probability that in these games “Khimik” will win the lot exactly twice.[[fill_choice_big-6]][[image-298]]","widgets":{"fill_choice_big-6":{"type":"fill_choice_big","options":["$0.375$","$0.3$","$0.25$"],"placeholder":0,"answer":0},"image-298":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/04/futbol.svg","width":"350"}},"step":1,"hints":["Look at the possible coin-toss outcomes: $$HHH, HHT, HTH, HTT, THH, THT, TTH, TTT$$","Let tails be the winning outcome. There are $3$ outcomes with exactly two tails out of $8.$ possible outcomes.","$$P(A)=\\frac{3}{8}=0.375$$"]},{"content":"One number is chosen at random from the set of positive integers from $15$ to $24$. What is the probability that it is divisible by $7?$[[fill_choice_big-7]][[image-339]]","widgets":{"fill_choice_big-7":{"type":"fill_choice_big","options":["$0.1$","$0.7$","$0.07$"],"placeholder":0,"answer":0},"image-339":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/11/obrazavr_angl_3_klass_papa_uchitel.svg","width":"350"}},"step":1,"hints":["In total, between $15$ and $24 $ there are $10 $ numbers. Among them, only one is divided into $7$:$21.$","$$P(A)=\\frac{1}{10}=0.1$$"]},{"content":"In the collection of mathematics question cards there are $45$ question cards, of which $18$ contains a question on the topic “Equations”. Find the probability that a student will not get a question on the topic “Equations” on a randomly selected exam question card.[[fill_choice_big-8]][[image-384]]","widgets":{"fill_choice_big-8":{"type":"fill_choice_big","options":["$0.4$","$0.6$","$0.45$"],"placeholder":0,"answer":1},"image-384":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/06/dumaet-nad-formulami-na-doske-formuly-doska.svg","width":"350"}},"step":1,"hints":["The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes.","Find the number of question cards that do not contain a question on the topic “Equations”: $$45-18=27$$","$$P(A)=\\frac{27}{45}=0.6$$"]},{"content":"The factory produces backpacks. On average, 95 backpacks from 100 have no hidden defects. Find the probability that the purchased backpack will be defective.[[fill_choice_big-9]][[image-433]]","widgets":{"fill_choice_big-9":{"type":"fill_choice_big","options":["$0.05$","$0.95$","$0.5$"],"placeholder":0,"answer":0},"image-433":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2022/07/bag-1.svg","width":"348"}},"hints":["Probability that the backpack will be defective: $$100-95=5$$","The probability of an event is the ratio of the number of favourable outcomes to the total number of equally likely outcomes: $$P(A)=\\frac{5}{100}$$"]},{"content":"In a random experiment, a fair coin is tossed twice. Find the probability that you will get tails twice.[[fill_choice_big-10]][[image-486]]","widgets":{"fill_choice_big-10":{"type":"fill_choice_big","options":["$0.2$","$0.5$","$0.25$"],"placeholder":0,"answer":2},"image-486":{"type":"image","url":"/wp-content/themes/Main/img/math-port/2023/11/svobodnoe5.svg","width":"350"}},"step":1,"hints":["List the possible events: $$HH, HT, TH, TT$$","Tails appears twice in one of the four outcomes:$$P(A)=\\frac{1}{4}=0.25$$"]}],"mix":1}