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NCERT Exemplar · Class 11 Mathematics Probability

43 questions · 43 still being checked

EXERCISE 16.3 31–43 (part 4 of 4)

  1. State whether the statements are True or False in each of the Exercises $\displaystyle 30$ to 36.

    Exercise 31

    The probability that a student will pass his examination is 0.73\displaystyle 0.73, the probability of the student getting a compartment is 0.13\displaystyle 0.13, and the probability that the student will either pass or get compartment is 0.96.

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    NCERT’s answer
    False
    FalseA student cannot both pass and get a compartment, so the events are mutually exclusive.\[P(\text{pass}\cup\text{compartment}) = 0.73+0.13 = 0.86 \] \[0.86 \ne 0.96 \]

  2. Exercise 32

    The probabilities that a typist will make 0\displaystyle 0, 1\displaystyle 1, 2\displaystyle 2, 3\displaystyle 3, 4\displaystyle 4, 5\displaystyle 5 or more mistakes in typing a report are, respectively, 0.12\displaystyle 0.12, 0.25\displaystyle 0.25, 0.36\displaystyle 0.36, 0.14\displaystyle 0.14, 0.08\displaystyle 0.08, 0.11.

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    NCERT’s answer
    False
    FalseThe six outcomes are exhaustive and mutually exclusive, so their probabilities must total 1.\[0.12+0.25+0.36+0.14+0.08+0.11 = 1.06 \] \[1.06 \ne 1 \]

  3. Exercise 33

    If A and B are two candidates seeking admission in an engineering College. The probability that A is selected is .5 and the probability that both A and B are selected is at most .3. Is it possible that the probability of B getting selected is 0.7\displaystyle 0.7?

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    NCERT’s answer
    True
    True. \(\displaystyle P(A\cup B)\le 1\) only forces \(\displaystyle P(A\cap B)\ge 0.2\), which is compatible with \(\displaystyle P(A\cap B)\le 0.3\). \[P(A\cup B) = P(A)+P(B)-P(A\cap B) = 0.5+0.7-P(A\cap B) \le 1 \] \[P(A\cap B) \ge 0.2 \] \[0.2 \le P(A\cap B) \le 0.3 \] Any value in this range, e.g. \(\displaystyle 0.25\), works.

  4. Exercise 34

    The probability of intersection of two events A and B is always less than or equal to those favourable to the event A.

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    NCERT’s answer
    True
    True. Every outcome of \(\displaystyle A\cap B\) is an outcome of \(\displaystyle A\). \[A\cap B \subseteq A \] \[P(A\cap B) \le P(A) \]

  5. Exercise 35

    The probability of an occurrence of event A is .7 and that of the occurrence of event B is .3 and the probability of occurrence of both is .4.

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    NCERT’s answer
    False
    False. \(\displaystyle A\cap B\subseteq B\), so \(\displaystyle P(A\cap B)\) cannot exceed \(\displaystyle P(B)\). \[P(A\cap B) = 0.4 > 0.3 = P(B) \]

  6. Exercise 36

    The sum of probabilities of two students getting distinction in their final examinations is 1.2.

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    NCERT’s answer
    True
    True. The two events are not mutually exclusive, so their probabilities can add to more than \(\displaystyle 1\); each only has to lie in \(\displaystyle [0,1]\). \[P(A) = P(B) = 0.6 \Rightarrow P(A)+P(B) = 1.2 \] \[P(A\cap B) = 0.6\times 0.6 = 0.36 \quad \text{(independent)} \] \[P(A\cup B) = 1.2 - 0.36 = 0.84 \le 1 \]
  7. Fill in the blanks in the Exercises $\displaystyle 37$ to 41.

    Exercise 37

    The probability that the home team will win an upcoming football game is 0.77\displaystyle 0.77, the probability that it will tie the game is 0.08\displaystyle 0.08, and the probability that it will lose the game is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    0.$\displaystyle 15$
    $\displaystyle 0.15$ \[P(\text{win}) + P(\text{tie}) + P(\text{lose}) = 1 \] \[P(\text{lose}) = 1 - 0.77 - 0.08 = 0.15 \]

  8. Exercise 38

    If e1,e2,e3,e4\displaystyle e_1, e_2, e_3, e_4 are the four elementary outcomes in a sample space and P(e1)=\displaystyle \mathrm{P}\left(e_1\right)= .1, P(e2)=.5,P(e3)=.1\displaystyle \mathrm{P}\left(e_2\right)=.5, \mathrm{P}\left(e_3\right)=.1, then the probability of e4\displaystyle e_4 is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    0.$\displaystyle 3$
    $\displaystyle 0.3$ \[P(e_1)+P(e_2)+P(e_3)+P(e_4) = 1 \] \[P(e_4) = 1 - (0.1+0.5+0.1) = 0.3 \]

  9. Exercise 39

    Let S={1,2,3,4,5,6}\displaystyle \mathrm{S}=\{1,2,3,4,5,6\} and E={1,3,5}\displaystyle \mathrm{E}=\{1,3,5\}, then E‾\displaystyle \overline{\mathrm{E}} is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    \(\displaystyle \bar{E}=\{2,4,6\}\)
    \(\displaystyle \{2,4,6\}\) \[\overline{E} = S - E = \{1,2,3,4,5,6\} - \{1,3,5\} = \{2,4,6\} \]

  10. Exercise 40

    If A and B are two events associated with a random experiment such that P(A)=0.3,P(B)=0.2\displaystyle \mathrm{P}(\mathrm{A})=0.3, \mathrm{P}(\mathrm{B})=0.2 and P(A∩B)=0.1\displaystyle \mathrm{P}(\mathrm{A} \cap \mathrm{B})=0.1, then the value of P(A∩B‾)\displaystyle \mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}}) is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    0.$\displaystyle 2$
    $\displaystyle 0.2$ \[P(A\cap \overline{B}) = P(A) - P(A\cap B) \] \[= 0.3 - 0.1 = 0.2 \]

  11. Exercise 41

    The probability of happening of an event A is 0.5\displaystyle 0.5 and that of B is 0.3. If A and B are mutually exclusive events, then the probability of neither A nor B is ____\displaystyle \_\_\_\_.

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    NCERT’s answer
    0.$\displaystyle 2$
    $\displaystyle 0.2$\[P(A \cup B) = P(A) + P(B) \quad \text{(mutually exclusive)} \] \[P(A \cup B) = 0.5 + 0.3 = 0.8 \] \[P(\text{neither } A \text{ nor } B) = P\big((A \cup B)'\big) = 1 - P(A \cup B) \] \[= 1 - 0.8 = 0.2 \]Answer: \(\displaystyle 0.2\)

  12. Exercise 42

    Match the proposed probability under Column C1\displaystyle \mathrm{C}_1 with the appropriate written description under column C2\displaystyle \mathrm{C}_2 :
    C1\displaystyle \mathbf{C}_{\mathbf{1}}C2\displaystyle \mathbf{C}_{\mathbf{2}}
    ProbabilityWritten Description
    (a) 0.95\displaystyle 0.95(i) An incorrect assignment
    (b) 0.02\displaystyle 0.02(ii) No chance of happening
    (c) −0.3\displaystyle -0.3(iii) As much chance of happening as not.
    (d) 0.5\displaystyle 0.5(iv) Very likely to happen
    (e) 0\displaystyle 0(v) Very little chance of happening

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    NCERT’s answer
    (a)
    ↔ (iv) (b) ↔ (v) (c) ↔ (i) (d) ↔ (iii) (e) ↔ (ii)
    (a)-(iv), (b)-(v), (c)-(i), (d)-(iii), (e)-(ii)\[0.95 \text{ is close to } 1 \;\Rightarrow\; \text{very likely} \quad \text{(a)-(iv)} \] \[0.02 \text{ is close to } 0 \;\Rightarrow\; \text{very little chance} \quad \text{(b)-(v)} \] \[-0.3 < 0 \;\Rightarrow\; \text{not a probability, an incorrect assignment} \quad \text{(c)-(i)} \] \[0.5 = \tfrac{1}{2} \;\Rightarrow\; \text{as much chance of happening as not} \quad \text{(d)-(iii)} \] \[P(E) = 0 \;\Rightarrow\; \text{no chance of happening} \quad \text{(e)-(ii)} \]Answer: (a)-(iv), (b)-(v), (c)-(i), (d)-(iii), (e)-(ii)

  13. Exercise 43

    Match the following
    (a) If E1\displaystyle \mathrm{E}_1 and E2\displaystyle \mathrm{E}_2 are the two mutually exclusive events(i) E1∩E2=E1\displaystyle \mathrm{E}_1 \cap \mathrm{E}_2=\mathrm{E}_1
    (b) If E1\displaystyle \mathrm{E}_1 and E2\displaystyle \mathrm{E}_2 are the mutually exclusive and exhaustive events(ii) (E1−E2)∪(E1∩E2)=E1\displaystyle \left(\mathrm{E}_1-\mathrm{E}_2\right) \cup\left(\mathrm{E}_1 \cap \mathrm{E}_2\right)=\mathrm{E}_1
    (c) If E1\displaystyle \mathrm{E}_1 and E2\displaystyle \mathrm{E}_2 have common outcomes, then(iii) E1∩E2=ϕ,E1∪E2=S\displaystyle \mathrm{E}_1 \cap \mathrm{E}_2=\phi, \mathrm{E}_1 \cup \mathrm{E}_2=\mathrm{S}
    (d) If E1\displaystyle \mathrm{E}_1 and E2\displaystyle \mathrm{E}_2 are two events such that E1⊂E2\displaystyle \mathrm{E}_1 \subset \mathrm{E}_2(iv) E1∩E2=ϕ\displaystyle \mathrm{E}_1 \cap \mathrm{E}_2=\phi

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    NCERT’s answer
    (a)
    ↔ (iv) (b) ↔ (iii) (c) ↔ (ii) (d) ↔ (i)
    (a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)\[\text{(a) mutually exclusive: no common outcome} \;\Rightarrow\; E_1 \cap E_2 = \phi \quad \text{(iv)} \] \[\text{(b) also exhaustive: together they fill } S \;\Rightarrow\; E_1 \cap E_2 = \phi,\; E_1 \cup E_2 = S \quad \text{(iii)} \] \[\text{(c) common outcomes: } E_1 \text{ splits into two parts} \;\Rightarrow\; (E_1 - E_2) \cup (E_1 \cap E_2) = E_1 \quad \text{(ii)} \] \[\text{(d) } E_1 \subset E_2 \;\Rightarrow\; E_1 \cap E_2 = E_1 \quad \text{(i)} \]Answer: (a)-(iv), (b)-(iii), (c)-(ii), (d)-(i)