Probability practice questions

Probability questions

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1. 1ProbabilityFoundation

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

   • A151/5
   • B454/5
   • C353/5
   • D121/2
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   Answer: 121/2

   There are 5 counters that are not blue out of 10 total. That is 510, which simplifies to 12.There are 5 counters that are not blue out of 10 total. That is 5/10, which simplifies to 1/2.

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2. 2ProbabilityFoundation

   A spinner has 8 equal sections, 3 of which are shaded. What is the probability of landing on an unshaded section?A spinner has 8 equal sections, 3 of which are shaded. What is the probability of landing on an unshaded section?

   • A535/3
   • B383/8
   • C585/8
   • D181/8
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   Answer: 585/8

   There are 8 - 3 = 5 unshaded sections out of 8 equally likely outcomes, so the probability is 58.There are 8 - 3 = 5 unshaded sections out of 8 equally likely outcomes, so the probability is 5/8.

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3. 3ProbabilityStandard

   A drawer holds 4 blue socks and 6 gray socks. One sock is taken out and kept. What is the probability that the first two socks taken are both gray?A drawer holds 4 blue socks and 6 gray socks. One sock is taken out and kept. What is the probability that the first two socks taken are both gray?

   • A9259/25
   • B353/5
   • C131/3
   • D252/5
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   Answer: 131/3

   The first gray sock has probability 610. With that sock removed, 5 of the remaining 9 are gray, so the second has probability 59. Multiplying gives 3090 = 13.The first gray sock has probability 6/10. With that sock removed, 5 of the remaining 9 are gray, so the second has probability 5/9. Multiplying gives 30/90 = 1/3.

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4. 4ProbabilityStandard

   A box holds 5 red, 3 green and 2 blue tokens. What is the probability that a token drawn at random is not red?A box holds 5 red, 3 green and 2 blue tokens. What is the probability that a token drawn at random is not red?

   • A121/2
   • B252/5
   • C3103/10
   • D5105/10
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   Answer: 121/2

   Five of the ten tokens are not red, so the probability is 510 = 12. Working with the complement is quicker than adding the green and blue cases separately, though both give the same answer.Five of the ten tokens are not red, so the probability is 5/10 = 1/2. Working with the complement is quicker than adding the green and blue cases separately, though both give the same answer.

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5. 5ProbabilityStandard

   A box holds 5 red counters, 7 blue counters and 8 green counters. One counter is drawn at random. What is the probability that it is not green?A box holds 5 red counters, 7 blue counters and 8 green counters. One counter is drawn at random. What is the probability that it is not green?

   • A252/5
   • B151/5
   • C353/5
   • D7207/20
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   Answer: 353/5

   There are 20 counters in all and 12 that are not green, so the probability is 1220 = 35. The choice 25 is the probability that it is green, which is what you get by answering the question that was not asked.There are 20 counters in all and 12 that are not green, so the probability is 12/20 = 3/5. The choice 2/5 is the probability that it is green, which is what you get by answering the question that was not asked.

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6. 6ProbabilityStandard

   Two fair six-sided dice are rolled. What is the probability that the two numbers total 9?Two fair six-sided dice are rolled. What is the probability that the two numbers total 9?

   • A161/6
   • B191/9
   • C4124/12
   • D1121/12
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   Answer: 191/9

   There are 36 equally likely outcomes and four give a total of 9: 3 and 6, 4 and 5, 5 and 4, 6 and 3. So the probability is 436 = 19. Counting 3 and 6 as the same outcome as 6 and 3 loses two of the four.There are 36 equally likely outcomes and four give a total of 9: 3 and 6, 4 and 5, 5 and 4, 6 and 3. So the probability is 4/36 = 1/9. Counting 3 and 6 as the same outcome as 6 and 3 loses two of the four.

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7. 7ProbabilityStandard

   A spinner has 8 equal sectors numbered 1 to 8. What is the probability of landing on a prime number?A spinner has 8 equal sectors numbered 1 to 8. What is the probability of landing on a prime number?

   • A383/8
   • B585/8
   • C141/4
   • D121/2
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   Answer: 121/2

   The primes from 1 to 8 are 2, 3, 5 and 7 — four of the eight, so the probability is 12. Counting 1 as prime gives 58; it is not, because a prime has exactly two distinct factors.The primes from 1 to 8 are 2, 3, 5 and 7 — four of the eight, so the probability is 1/2. Counting 1 as prime gives 5/8; it is not, because a prime has exactly two distinct factors.

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8. 8ProbabilityStandard

   Two fair coins are tossed. What is the probability of getting at least one head?Two fair coins are tossed. What is the probability of getting at least one head?

   • A121/2
   • B141/4
   • C232/3
   • D343/4
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   Answer: 343/4

   The four equally likely outcomes are HH, HT, TH and TT, and three of them contain a head, so the probability is 34. It is quicker as 1 minus the probability of no heads: 1 - 14.The four equally likely outcomes are HH, HT, TH and TT, and three of them contain a head, so the probability is 3/4. It is quicker as 1 minus the probability of no heads: 1 - 1/4.

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9. 9ProbabilityStandard

   Two fair six-sided dice are rolled. What is the probability that the total is 7?Two fair six-sided dice are rolled. What is the probability that the total is 7?

   • A1121/12
   • B161/6
   • C7367/36
   • D191/9
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   Answer: 161/6

   Six of the 36 outcomes total 7 — (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — so the probability is 636 = 16. Seven is the likeliest total, which is worth remembering.Six of the 36 outcomes total 7 — (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) — so the probability is 6/36 = 1/6. Seven is the likeliest total, which is worth remembering.

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10. 10ProbabilityStandard

    Two fair six-sided dice are rolled. What is the probability that the total is 5?Two fair six-sided dice are rolled. What is the probability that the total is 5?

    • A1121/12
    • B191/9
    • C5365/36
    • D161/6
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    Answer: 191/9

    Four of the 36 outcomes total 5 — (1,4), (2,3), (3,2) and (4,1) — so the probability is 436 = 19. Counting (1,4) and (4,1) as one outcome would halve it wrongly.Four of the 36 outcomes total 5 — (1,4), (2,3), (3,2) and (4,1) — so the probability is 4/36 = 1/9. Counting (1,4) and (4,1) as one outcome would halve it wrongly.

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11. 11ProbabilityChallenge

    Two fair coins are tossed. What is the probability that at least one shows heads?Two fair coins are tossed. What is the probability that at least one shows heads?

    • A343/4
    • B232/3
    • C121/2
    • D141/4
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    Answer: 343/4

    The four equally likely outcomes are HH, HT, TH, TT. Three contain at least one head. Working with the complement is faster: the only outcome without a head is TT, so 1 - 14 = 34.The four equally likely outcomes are HH, HT, TH, TT. Three contain at least one head. Working with the complement is faster: the only outcome without a head is TT, so 1 - 1/4 = 3/4.

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12. 12ProbabilityChallenge

    A bag holds 4 white and 6 black marbles. Two are drawn without replacement. What is the probability both are white?A bag holds 4 white and 6 black marbles. Two are drawn without replacement. What is the probability both are white?

    • A151/5
    • B2152/15
    • C8458/45
    • D4254/25
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    Answer: 2152/15

    The first is white with probability 410. With it removed, 3 of the remaining 9 are white, so the second is 39. Multiplying gives 1290 = 215.The first is white with probability 4/10. With it removed, 3 of the remaining 9 are white, so the second is 3/9. Multiplying gives 12/90 = 2/15.

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13. 13ProbabilityChallenge

    A bag holds 4 white marbles and 6 black ones. Two marbles are drawn without replacement. What is the probability that both are white?A bag holds 4 white marbles and 6 black ones. Two marbles are drawn without replacement. What is the probability that both are white?

    • A4254/25
    • B2152/15
    • C151/5
    • D252/5
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    Answer: 2152/15

    The first draw is white with probability 410. The second is then drawn from 9 marbles of which 3 are white, so 39. Multiplying gives 1290 = 215. Using 410 twice gives 425 and quietly puts the first marble back.The first draw is white with probability 4/10. The second is then drawn from 9 marbles of which 3 are white, so 3/9. Multiplying gives 12/90 = 2/15. Using 4/10 twice gives 4/25 and quietly puts the first marble back.

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