Probability I

28 Grade 10 Essential Mathematics questions on Probability I, with worked marking schemes. 28 questions are available on this topic in full.

1. Question 1

   1 mark

   Records kept at a matatu stage show that the probability that the 7.00 a.m. matatu arrives late is 0.35. What is the probability that it does not arrive late?

   A. 0.5

   B. 1.35

   C. 0.35

   D. 0.65

   (1 mark)

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2. Question 2

   3 marks

   A poultry farmer in Kakamega recorded that out of 250 chicks brooded last season, 20 died before they were four weeks old.

   (a) Determine the experimental probability that a chick dies before four weeks.

   (b) Suggest one reason why this probability may be different next season.

   (3 marks)

   Marking scheme

   (a) Probability = number of chicks that died / total number brooded = $\frac{20}{250}$ (1 mark) = $\frac{2}{25}$ or 0.08 (1 mark). (b) Any one valid reason (1 mark): better or worse brooding conditions; different weather or temperature; a disease outbreak; improved feeding or vaccination; a different breed of chick; a different number of chicks brooded.

3. Question 3

   6 marks

   A fruit vendor in Nakuru sold 100 fruits one morning, made up of 40 mangoes, 25 avocados, 20 bananas and 15 oranges. One of these sales is selected at random for a survey.

   (a) Determine the probability that the sale selected was a mango.

   (b) Determine the probability that the sale selected was not an orange.

   (c) The vendor claims that the probability of selling a banana that morning was $\frac{1}{5}$. Explain whether the records support this claim.

   (6 marks)

   Marking scheme

   (a) Probability = $\frac{40}{100}$ (1 mark) = $\frac{2}{5}$ or 0.4 (1 mark). (b) Sales that were not oranges = 100 - 15 = 85 (1 mark); probability = $\frac{85}{100}=\frac{17}{20}$ or 0.85 (1 mark). Accept $1-\frac{15}{100}$. (c) Probability of a banana = $\frac{20}{100}=\frac{1}{5}$ (1 mark); therefore the records support the claim, since the two values are equal (1 mark).

4. Question 4

   1 mark

   Four events have the following probabilities. Which event is the most likely to occur?

   A. An event with probability 58%

   B. An event with probability $\frac{1}{2}$

   C. An event with probability 0.55

   D. An event with probability $\frac{3}{5}$

   (1 mark)

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5. Question 5

   1 mark

   In a factory, the probability that a bulb picked at random is faulty is 0.02. Which word best describes the event that a bulb picked at random is faulty?

   A. Impossible

   B. Very likely

   C. Very unlikely

   D. Certain

   (1 mark)

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6. Question 6

   3 marks

   Two fair coins are tossed together once.

   (a) Outline all the possible outcomes.

   (b) Determine the probability of obtaining exactly one head.

   (3 marks)

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7. Question 7

   1 mark

   A learner tosses a bottle top 200 times and it lands facing up 72 times. What is the experimental probability that the bottle top lands facing up?

   A. 0.64

   B. 0.56

   C. 0.36

   D. 0.5

   (1 mark)

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8. Question 8

   3 marks

   Explain, giving one example, how insurance companies in Kenya make use of probability.

   (3 marks)

   Marking scheme

   They use past records to estimate the probability (chance) that a particular loss or claim will occur (1 mark). They use this estimate to decide how much premium each client should pay, so that the money collected covers the expected claims (1 mark). Valid example (1 mark): estimating the probability that a matatu is involved in an accident in a year before fixing its motor insurance premium; or estimating the chance of crop failure before pricing crop insurance for a maize farmer.

9. Question 9

   1 mark

   The probability that an egg placed in an incubator hatches successfully is 0.9. A poultry farmer places 300 eggs in the incubator. How many eggs are expected to hatch?

   A. 270

   B. 2700

   C. 27

   D. 30

   (1 mark)

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10. Question 10

    1 mark

    A weather station classifies each day as sunny, cloudy or rainy. For a certain day it gives the probability of sunny weather as 0.5 and of cloudy weather as 0.3. What is the probability of rainy weather on that day?

    A. 0.7

    B. 0.5

    C. 0.2

    D. 0.8

    (1 mark)

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11. Question 11

    4 marks

    A boda boda rider keeps daily records. On any working day, the probability that he carries fewer than 10 customers is 0.15, the probability that he carries from 10 to 20 customers is 0.5, and the probability that he carries more than 20 customers is p.

    (a) Determine the value of p.

    (b) Determine the probability that on a given day he carries 10 or more customers.

    (4 marks)

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12. Question 12

    1 mark

    A pencil case contains only red pens and blue pens. One pen is picked from the case at random. What is the probability that the pen picked is green?

    A. 0

    B. $\frac{1}{3}$

    C. 1

    D. $\frac{1}{2}$

    (1 mark)

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13. Question 13

    2 marks

    A learner tosses a fair coin five times and obtains a tail each time. He insists that the sixth toss must give a head. Explain why his reasoning is wrong.

    (2 marks)

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14. Question 14

    1 mark

    A learner tosses a fair coin 10 times and obtains 7 heads instead of the expected 5. Which statement best explains this result?

    A. The next 10 tosses will give only 3 heads to balance the results

    B. The coin must be biased towards heads

    C. The theoretical probability of a head is not $\frac{1}{2}$

    D. The number of tosses is small, so the experimental probability may differ from the theoretical one

    (1 mark)

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15. Question 15

    5 marks

    In a Grade 10 class of 40 learners, the probability that a learner picked at random walks to school is $\frac{2}{5}$.

    (a) Determine the number of learners who walk to school.

    (b) Determine the probability that a learner picked at random does not walk to school.

    (c) State, giving a reason, whether picking a learner who walks to school is more likely or less likely than picking one who does not.

    (5 marks)

    Marking scheme

    (a) Number who walk = $\frac{2}{5}x$ 40 (1 mark) = 16 learners (1 mark). (b) Probability = $1-\frac{2}{5}=\frac{3}{5}$ (or $\frac{24}{40}$) (1 mark). (c) Less likely (1 mark), because $\frac{2}{5}$ is smaller than $\frac{3}{5}$, i.e. fewer learners (16) walk than do not walk (24) (1 mark).

16. Question 16

    1 mark

    A learner states that the probability that it will rain in Kisii tomorrow is 1.4. Which statement explains why 1.4 cannot be a probability?

    A. A probability must always be a whole number

    B. A probability must always be written as a percentage

    C. A probability must be greater than 1 for an event that is likely

    D. A probability must lie between 0 and 1 inclusive

    (1 mark)

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17. Question 17

    2 marks

    Distinguish between theoretical probability and experimental probability.

    (2 marks)

    Marking scheme

    Theoretical probability is worked out by reasoning about equally likely outcomes, without carrying out the experiment, i.e. favourable outcomes divided by total possible outcomes (1 mark). Experimental probability is obtained from the results of an actual experiment or from records, i.e. the number of times the event occurred divided by the number of trials carried out (1 mark).

18. Question 18

    2 marks

    A tray holds 30 eggs, 6 of which are cracked. One egg is picked from the tray at random. Determine the probability that the egg picked is not cracked.

    (2 marks)

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19. Question 19

    1 mark

    Which one of the following events has a probability of 1?

    A. A learner picked at random from a Grade 10 class being a Grade 10 learner

    B. A dairy cow on a farm giving birth to twin calves

    C. Tossing a fair coin once and obtaining a head

    D. Rain falling in Nairobi during the coming week

    (1 mark)

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20. Question 20

    1 mark

    A fair die with faces numbered 1 to 6 is rolled once. What is the probability of obtaining a number greater than 4?

    A. $\frac{1}{6}$

    B. $\frac{1}{2}$

    C. $\frac{2}{3}$

    D. $\frac{1}{3}$

    (1 mark)

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21. Question 21

    1 mark

    At a school fundraiser, 500 raffle tickets are sold and one winning ticket is drawn at random. A parent buys 20 of the tickets. What is the probability that the parent wins?

    A. $\frac{1}{24}$

    B. $\frac{1}{500}$

    C. $\frac{1}{20}$

    D. $\frac{1}{25}$

    (1 mark)

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22. Question 22

    1 mark

    A tin on a farm contains 8 maize seeds and 12 bean seeds, all of the same size. One seed is picked from the tin at random. What is the probability that the seed picked is a maize seed?

    A. $\frac{2}{3}$

    B. $\frac{2}{5}$

    C. $\frac{1}{8}$

    D. $\frac{3}{5}$

    (1 mark)

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23. Question 23

    4 marks

    A weather forecast gives the probability of rain in Machakos tomorrow as 0.2.

    (a) State what this value tells a farmer about tomorrow's weather.

    (b) Determine the probability that it does not rain.

    (c) Explain why it may still rain tomorrow even though this probability is low.

    (4 marks)

    Marking scheme

    (a) Rain is unlikely; on about 1 in every 5 days with such conditions, rain falls (1 mark). (b) 1 - 0.2 = 0.8 (1 mark). (c) A probability shows the chance of an event, not a guarantee that it will or will not happen (1 mark); the forecast is based on past records and present conditions, and weather conditions can change (1 mark).

24. Question 24

    2 marks

    A letter is chosen at random from the letters of the word KISUMU. Determine the probability that the letter chosen is U.

    (2 marks)

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25. Question 25

    1 mark

    A fair die with faces numbered 1 to 6 is rolled once. Which event has the same probability as obtaining an odd number?

    A. Obtaining a number greater than 2

    B. Obtaining a number greater than 4

    C. Obtaining the number 5

    D. Obtaining a number less than 4

    (1 mark)

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26. Question 26

    4 marks

    A fair die with faces numbered 1 to 6 is rolled once.

    (a) Determine the probability of obtaining a prime number.

    (b) Determine the probability of obtaining a number that is not prime.

    (c) State the sum of the two probabilities obtained in (a) and (b).

    (4 marks)

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27. Question 27

    3 marks

    The probability that a certified maize seed germinates is 0.85. A farmer plants 400 such seeds. Determine the expected number of seeds that will not germinate.

    (3 marks)

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28. Question 28

    5 marks

    A crate in a school canteen contains 24 bottles: 10 bottles of soda, 8 bottles of juice and 6 bottles of drinking water. A learner picks one bottle at random. Determine the probability that the bottle picked is (a) juice, (b) not water, (c) either soda or water.

    (5 marks)

    Marking scheme

    (a) $\frac{8}{24}=\frac{1}{3}$ (1 mark). (b) Bottles that are not water = 24 - 6 = 18 (1 mark); probability = $\frac{18}{24}=\frac{3}{4}$ (1 mark). Accept $1-\frac{6}{24}=\frac{3}{4}$. (c) Soda or water = 10 + 6 = 16 (1 mark); probability = $\frac{16}{24}=\frac{2}{3}$ (1 mark).

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