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NCERT Exemplar · Class 10 Mathematics Statistics and Probability

96 questions · 96 still being checked

EXERCISE 13.1 11–20 (part 2 of 11)

  1. The number of atheletes who completed the race in less then $\displaystyle 14.6$ seconds is :

    Exercise 11

    Consider the following distribution :
    Marks obtainedNumber of students
    More than or equal to 0\displaystyle 063\displaystyle 63
    More than or equal to 10\displaystyle 1058\displaystyle 58
    More than or equal to 20\displaystyle 2055\displaystyle 55
    More than or equal to 30\displaystyle 3051\displaystyle 51
    More than or equal to 40\displaystyle 4048\displaystyle 48
    More than or equal to 50\displaystyle 5042\displaystyle 42
    the frequency of the class 30\displaystyle 30-40\displaystyle 40 is
    (A)
    3\displaystyle 3 (B) 4\displaystyle 4 (C) 48\displaystyle 48
    (D)
    51\displaystyle 51

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    NCERT’s answer
    (A)
    (A) \(\displaystyle 3\) \[f_{30\text{-}40} = (\text{cf} \ge 30) - (\text{cf} \ge 40) = 51-48=3 \] A "more than" table gives each class frequency as the drop to the next row.

  2. Exercise 12

    If an event cannot occur, then its probability is
    (A)
    1\displaystyle 1 (B) 34\displaystyle \frac{3}{4}
    (C)
    12\displaystyle \frac{1}{2}

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    NCERT’s answer
    (D)
    (D) \(\displaystyle 0\) \[0 \le P(E) \le 1 \] An event that cannot occur is the impossible event, at the lower bound of this range.

  3. Exercise 13

    Which of the following cannot be the probability of an event?
    (A)
    13\displaystyle \frac{1}{3}
    (B)
    0.1\displaystyle 1
    (C)
    3\displaystyle 3% (D) 1716\displaystyle \frac{17}{16}

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    NCERT’s answer
    (D)
    (D) \(\displaystyle \dfrac{17}{16}\) \[0 \le P(E) \le 1,\qquad \frac{17}{16}=1.0625 \] \(\displaystyle 1.0625\) lies outside \(\displaystyle [0,1]\), so it cannot be a probability.

  4. Exercise 14

    An event is very unlikely to happen. Its probability is closest to
    (A)
    0.0001\displaystyle 0001
    (B)
    0.001\displaystyle 001
    (C)
    0.01\displaystyle 01
    (D)
    0.1\displaystyle 1

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    NCERT’s answer
    (A)
    (A) \(\displaystyle 0.0001\) \[0 \le P(E) \le 1 \] \(\displaystyle 0\) marks an impossible event and \(\displaystyle 1\) a certain one; an event very unlikely to happen sits nearest the impossible end, closest to \(\displaystyle 0.0001\).

  5. Exercise 15

    If the probability of an event is p\displaystyle p, the probability of its complementary event will be
    (A)
    p−1\displaystyle p-1
    (B)
    p\displaystyle p (C) 1−p\displaystyle 1-p
    (D)
    1−1p\displaystyle 1-\frac{1}{p}

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    NCERT’s answer
    (C)
    (C) \(\displaystyle 1-p\)An event and its complement partition the sample space, so their probabilities add to \(\displaystyle 1\): \[P(E) + P(\bar{E}) = 1 \] \[P(\bar{E}) = 1 - p \]

  6. Exercise 16

    The probability expressed as a percentage of a particular occurrence can never be
    (A)
    less than 100\displaystyle 100
    (B)
    less than 0\displaystyle 0
    (C)
    greater than 1\displaystyle 1
    (D)
    anything but a whole number

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    NCERT’s answer
    (B)
    (B) less than \(\displaystyle 0\)Since \(\displaystyle 0 \le P(E) \le 1\), as a percentage \(\displaystyle P(E)\) lies in \(\displaystyle [0,100]\) — never negative, but it can be less than \(\displaystyle 100\), greater than \(\displaystyle 1\), or a fraction.

  7. Exercise 17

    If P(A)\displaystyle \mathrm{P}(\mathrm{A}) denotes the probability of an event A, then
    (A)
    P(A)<0\displaystyle \mathrm{P}(\mathrm{A})<0
    (B)
    P(A)>1\displaystyle \mathrm{P}(\mathrm{A})>1
    (C)
    0≤P(A)≤1\displaystyle 0 \leq \mathrm{P}(\mathrm{A}) \leq 1
    (D)
    −1≤P(A)≤1\displaystyle -1 \leq \mathrm{P}(\mathrm{A}) \leq 1

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    NCERT’s answer
    (C)
    (C) \(\displaystyle 0 \leq \mathrm{P}(\mathrm{A}) \leq 1\)Probability is the ratio of favourable to total equally likely outcomes, so it is bounded: \[0 \le \mathrm{P}(\mathrm{A}) \le 1 \]

  8. Exercise 18

    A card is selected from a deck of 52\displaystyle 52 cards. The probability of its being a red face card is
    (A)
    326\displaystyle \frac{3}{26}
    (B)
    313\displaystyle \frac{3}{13}
    (C)
    213\displaystyle \frac{2}{13}
    (D)
    12\displaystyle \frac{1}{2}

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    NCERT’s answer
    (A)
    (A) \(\displaystyle \frac{3}{26}\)Face cards are J, Q, K; the red suits are hearts and diamonds: \[\text{red face cards} = 2 \times 3 = 6 \] \[P(\text{red face card}) = \frac{6}{52} = \frac{3}{26} \]

  9. Exercise 19

    The probability that a non leap year selected at random will contain 53\displaystyle 53 sundays is
    (A)
    17\displaystyle \frac{1}{7}
    (B)
    27\displaystyle \frac{2}{7}
    (C)
    37\displaystyle \frac{3}{7}
    (D)
    57\displaystyle \frac{5}{7}

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    NCERT’s answer
    (A)
    (A) \(\displaystyle \frac{1}{7}\)A non-leap year has \(\displaystyle 365\) days: \[365 = 52 \times 7 + 1 \] The \(\displaystyle 52\) weeks give \(\displaystyle 52\) Sundays; the leftover day is equally likely to be any of the \(\displaystyle 7\) days: \[P(53 \text{ Sundays}) = \frac{1}{7} \]

  10. Exercise 20

    When a die is thrown, the probability of getting an odd number less than 3\displaystyle 3 is
    (A)
    16\displaystyle \frac{1}{6}
    (B)
    13\displaystyle \frac{1}{3}
    (C)
    12\displaystyle \frac{1}{2}

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    NCERT’s answer
    (A)
    (A) \(\displaystyle \frac{1}{6}\)Odd numbers on a die are \(\displaystyle 1,3,5\); of these, only \(\displaystyle 1\) is less than \(\displaystyle 3\): \[P(\text{odd and} < 3) = \frac{1}{6} \]