Probability Basics and Relative Frequency

1. 1 Write down [2 marks]

   A bag contains 6 red counters, 3 blue counters and 1 green counter. Kay takes a counter at random. Write down the probability that the counter is (a) blue, (b) not red.

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   Model answer

   (a) \(\dfrac{3}{10}\). (b) There are 4 counters that are not red, so \(\dfrac{4}{10} = \dfrac{2}{5}\).

   Mark scheme

   • (a) \(\dfrac{3}{10}\) or 0.3 — B1
   • (b) \(\dfrac{4}{10}\) or \(\dfrac{2}{5}\) or 0.4 — B1
2. 2 Work out [4 marks]

   Here are the probabilities of the scores on a biased dice. The probability of a 1 is 0.1, of a 2 is 0.15, of a 3 is 0.2, of a 4 is \(x\), of a 5 is 0.25 and of a 6 is 0.1. (a) Work out the value of \(x\). (2 marks) (b) Mel throws the dice 200 times. Work out an estimate for the number of times she gets a 5. (2 marks)

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   Model answer

   (a) The other probabilities add up to \(0.1 + 0.15 + 0.2 + 0.25 + 0.1 = 0.8\), so \(x = 1 - 0.8 = 0.2\). (b) \(0.25 \times 200 = 50\).

   Mark scheme

   • (a) \(0.1 + 0.15 + 0.2 + 0.25 + 0.1 = 0.8\) or \(1 - 0.8\) — M1
   • (a) \(0.2\) — A1
   • (b) \(0.25 \times 200\) — M1
   • (b) 50 — A1
3. 3 Explain [3 marks]

   Ali throws a dice 150 times. It lands on 6 a total of 45 times. (a) Work out the relative frequency of getting a 6. (1 mark) (b) Ali says, “The dice is fair.” Is Ali correct? Give a reason for your answer. (2 marks)

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   Model answer

   (a) \(\dfrac{45}{150} = 0.3\). (b) For a fair dice the probability of a 6 is \(\dfrac{1}{6}\), which is about 0.17. The relative frequency 0.3 is much larger, so the dice is probably not fair.

   Mark scheme

   • (a) \(0.3\) or \(\dfrac{3}{10}\) — B1
   • (b) Compares 0.3 with \(\dfrac{1}{6}\) or about 0.17 — M1
   • (b) Not fair, because the relative frequency is much bigger than for a fair dice — C1
4. 4 Work out [2 marks]

   The probability that a train is late is 0.08. Mia takes the train on 250 days. Work out an estimate for the number of days that the train is late.

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   Model answer

   \(0.08 \times 250 = 20\).

   Mark scheme

   • \(0.08 \times 250\) or \(\dfrac{8}{100} \times 250\) — M1
   • 20 — A1
5. 5 Work out [3 marks]

   Jo plays a game. The probability that she wins is 0.35. The probability that she draws is 0.25. (a) Work out the probability that she loses. (1 mark) Jo plays the game 40 times. (b) Work out an estimate for the number of times she wins. (2 marks)

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   Model answer

   (a) \(1 - 0.35 - 0.25 = 0.4\). (b) \(0.35 \times 40 = 14\).

   Mark scheme

   • (a) \(0.4\) — B1
   • (b) \(0.35 \times 40\) — M1
   • (b) 14 — A1
6. 6 Work out [3 marks]

   In a school, the probability that a student chosen at random is a girl is 0.6. The probability that a student plays chess is 0.3. The probability that a student is a girl and plays chess is 0.2. Work out the probability that a student chosen at random is a girl or plays chess.

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   Model answer

   \(P(\text{girl or chess}) = P(\text{girl}) + P(\text{chess}) - P(\text{girl and chess}) = 0.6 + 0.3 - 0.2 = 0.7\).

   Mark scheme

   • \(0.6 + 0.3\) — M1
   • \(0.6 + 0.3 - 0.2\) — M1
   • \(0.7\) — A1

Quick check

1. 1

   What is the probability of an impossible event?

   1. A\(1\)
   2. B\(0\)
   3. C\(\dfrac{1}{2}\)
   4. D\(-1\)
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   B: \(0\)

   An impossible event has probability 0.

2. 2

   A bag has 3 red, 5 blue and 2 green counters. One is taken at random. What is the probability it is blue?

   1. A\(\dfrac{1}{2}\)
   2. B\(\dfrac{1}{5}\)
   3. C\(\dfrac{1}{3}\)
   4. D\(\dfrac{3}{5}\)
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   A: \(\dfrac{1}{2}\)

   There are 10 counters and 5 are blue, so \(\dfrac{5}{10} = \dfrac{1}{2}\).

3. 3

   A bag has 3 red, 5 blue and 2 green counters. What is the probability that a counter taken at random is not green?

   1. A\(\dfrac{1}{5}\)
   2. B\(\dfrac{2}{5}\)
   3. C\(\dfrac{3}{10}\)
   4. D\(\dfrac{4}{5}\)
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   D: \(\dfrac{4}{5}\)

   \(1 - \dfrac{2}{10} = \dfrac{8}{10} = \dfrac{4}{5}\).

4. 4

   A dice is thrown 120 times and a 6 comes up 30 times. What is the relative frequency of a 6?

   1. A\(\dfrac{1}{6}\)
   2. B\(\dfrac{1}{30}\)
   3. C\(\dfrac{1}{4}\)
   4. D\(4\)
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   C: \(\dfrac{1}{4}\)

   \(\dfrac{30}{120} = \dfrac{1}{4}\).

5. 5

   A fair dice is thrown 600 times. How many sixes are expected?

   1. A\(60\)
   2. B\(100\)
   3. C\(120\)
   4. D\(600\)
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   B: \(100\)

   \(\dfrac{1}{6} \times 600 = 100\).

6. 6

   Two fair dice are thrown. What is the probability that the total score is 7?

   1. A\(\dfrac{1}{6}\)
   2. B\(\dfrac{1}{12}\)
   3. C\(\dfrac{7}{36}\)
   4. D\(\dfrac{1}{36}\)
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   A: \(\dfrac{1}{6}\)

   There are 6 ways to make 7 out of 36, so \(\dfrac{6}{36} = \dfrac{1}{6}\).

7. 7

   A bag has 3 red, 5 blue and 2 green counters. What is the probability of taking a red or a green counter?

   1. A\(\dfrac{3}{5}\)
   2. B\(\dfrac{1}{5}\)
   3. C\(\dfrac{3}{10}\)
   4. D\(\dfrac{1}{2}\)
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   D: \(\dfrac{1}{2}\)

   The events are mutually exclusive, so \(\dfrac{3}{10} + \dfrac{2}{10} = \dfrac{1}{2}\).

8. 8

   A spinner is spun 200 times and lands on red 74 times. How many reds would be expected if the probability of red were \(\dfrac{1}{4}\)?

   1. A\(74\)
   2. B\(25\)
   3. C\(50\)
   4. D\(200\)
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   C: \(50\)

   \(\dfrac{1}{4} \times 200 = 50\). The result of 74 suggests the spinner may be biased.

9. 9

   A card is taken from a pack of 52. What is the probability that it is a heart or a king?

   1. A\(\dfrac{17}{52}\)
   2. B\(\dfrac{4}{13}\)
   3. C\(\dfrac{1}{13}\)
   4. D\(\dfrac{5}{26}\)
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   B: \(\dfrac{4}{13}\)

   \(\dfrac{13}{52} + \dfrac{4}{52} - \dfrac{1}{52} = \dfrac{16}{52} = \dfrac{4}{13}\).