Probability: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. $P(A)=0.5$, $P(A\cap B)=0.2$: $P(B|A)$.
2. A die: P(even | number > 3).
3. $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{4}$, independent: $P(A\cap B)$.
4. Two coins: P(both heads | at least one head).
5. $P(A)=0.6$, $P(B)=0.3$, $P(A\cap B)=0.18$: independent?
6. Find $k$: $P(X)$ = $k$, $2k$, $3k$, $4k$.
7. Mean of X = 0, 1, 2 with $\frac{1}{4},\frac{1}{2},\frac{1}{4}$.
8. Independent, $P(A)=0.2$, $P(B)=0.5$: $P(A^{\prime }\cap B^{\prime })$.
9. One card: P(king | face card).
10. $P(E|F)$ when $E\cap F=\varphi$.

D. Short answer I (2 marks each)

1. $P(A)=0.4$, $P(B)=0.8$, $P(B|A)=0.6$: find $P(A|B)$ and $P(A\cup B)$.
2. A family has three children: P(all girls | at least one girl).
3. Two cards drawn without replacement: P(both aces).
4. Independent, $P(A)=0.3$, $P(B)=0.4$: P(exactly one occurs).
5. A speaks the truth in 60% of cases, B in 90%, independently: P(they contradict each other on a fact).
6. Probability distribution of the number of heads in two tosses of a coin.
7. Mean number of heads in three tosses.
8. Show that if A, B are independent then A and B′ are independent.

E. Short answer II (3 marks each)

1. Bag I: 3 red, 2 black; bag II: 2 red, 4 black. A bag is chosen at random and a red ball drawn: P(bag I).
2. Machines A, B, C make 25%, 35%, 40% of the output; 5%, 4%, 2% of their items are defective. A defective item: P(made by B).
3. Two cards drawn with replacement: distribution and mean of the number of aces.
4. $P(X=x)=kx$ for $x=1,2,3,4$: find $k$, $P(X\ge 3)$ and the mean.
5. A and B throw a die alternately, A first; the first to throw a 6 wins. Find P(A wins).
6. Independent, $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$: P(at least one) and P(neither).

F. Long answer (5 marks each)

1. Bag I has 1 white, 2 black, 3 red balls; bag II 2 white, 1 black, 1 red; bag III 4 white, 5 black, 3 red. A bag is chosen at random and two balls drawn: one white, one red. Find the probability that they came from bag II.
2. Two bulbs are drawn without replacement from 10 bulbs of which 3 are defective. Find the probability distribution of the number of defective bulbs, its mean and its variance.

G. HOTS (higher-order thinking)

1. Why are mutually exclusive events with positive probabilities never independent?
2. Why is Bayes' theorem called "reverse probability"?
3. What does $P(A|B)>P(A)$ tell you?
4. Why must the probabilities of a distribution add up to 1?

20 MCQs

1. $P(E|F)=$
   (a) $P(E)P(F)$ (b) $\frac{P(E\cup F)}{P(F)}$ (c) $\frac{P(E\cap F)}{P(F)}$ (d) $\frac{P(F)}{P(E)}$
2. $P(A)=0.4$, $P(B)=0.5$, independent: $P(A\cap B)=$
   (a) 0.2 (b) 0.9 (c) 0.1 (d) 0.45
3. $P(A|A)=$
   (a) 0 (b) $\frac{1}{2}$ (c) $P(A)$ (d) 1
4. A die: P(6 | even)
   (a) $\frac{1}{6}$ (b) $\frac{1}{3}$ (c) $\frac{1}{2}$ (d) 1
5. A, B mutually exclusive, $P(B)>0$: $P(A|B)=$
   (a) 1 (b) 0 (c) $P(A)$ (d) $P(B)$
6. $P(A)=\frac{1}{2}$, $P(B)=\frac{1}{3}$, $P(A\cap B)=\frac{1}{6}$: A and B are
   (a) mutually exclusive (b) complementary (c) equal (d) independent
7. Two coins: P(two heads | first is a head)
   (a) $\frac{1}{2}$ (b) $\frac{1}{4}$ (c) $\frac{1}{3}$ (d) 1
8. If $P(B)=0$, $P(A|B)$ is
   (a) 0 (b) 1 (c) not defined (d) $P(A)$
9. Mean of X = 1, 2, 3 with 0.2, 0.5, 0.3
   (a) 2 (b) 2.5 (c) 1.9 (d) 2.1
10. $k$ if $P(X)$: 0.1, $k$, 0.2, $2k$, 0.1
    (a) 0.1 (b) 0.2 (c) 0.3 (d) 0.4
11. Independent, $P(A)=0.3$, $P(B)=0.6$: $P(A\cup B)$
    (a) 0.9 (b) 0.18 (c) 0.72 (d) 0.81
12. $P(E_1)=0.4$, $P(E_2)=0.6$, $P(A|E_1)=0.5$, $P(A|E_2)=0.25$: $P(A)$
    (a) 0.35 (b) 0.75 (c) 0.5 (d) 0.25
13. In Q12, $P(E_1|A)=$
    (a) $\frac{3}{7}$ (b) $\frac{4}{7}$ (c) $\frac{1}{2}$ (d) $\frac{2}{7}$
14. At least one head in 3 tosses
    (a) $\frac{1}{8}$ (b) $\frac{1}{2}$ (c) $\frac{7}{8}$ (d) $\frac{3}{8}$
15. If $A\subset B$, $P(B|A)=$
    (a) 0 (b) $P(A)$ (c) $P(B)$ (d) 1
16. Independent, $P(A)=P(B)=\frac{1}{2}$: $P(A^{\prime }\cap B^{\prime })$
    (a) $\frac{1}{4}$ (b) $\frac{1}{2}$ (c) $\frac{3}{4}$ (d) 0
17. A, B independent: $P(A|B)=$
    (a) $P(A)$ (b) $P(B)$ (c) 1 (d) 0
18. Two dice: P(sum 7 | first die shows 3)
    (a) $\frac{1}{36}$ (b) $\frac{1}{2}$ (c) $\frac{1}{3}$ (d) $\frac{1}{6}$
19. A speaks the truth with $\frac{3}{4}$ and reports a head: P(head)
    (a) $\frac{1}{2}$ (b) $\frac{3}{4}$ (c) $\frac{1}{4}$ (d) 1
20. $\sum P(X=x_i)=$
    (a) 0 (b) the mean (c) 1 (d) any number

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: $P(A)=0.5$, $P(B)=0.4$, $P(A\cap B)=0.2$ make A and B independent. R: Independent events satisfy $P(A\cap B)=P(A)P(B)$.
2. A: $P(A|B)=1$ when $B\subset A$ and $P(B)\ne 0$. R: $P(A\cap B)=P(B)$ when $B\subset A$.
3. A: Two mutually exclusive events with positive probabilities are independent. R: For mutually exclusive events $P(A\cap B)=0$.
4. A: $P(E^{\prime }|F)=1-P(E|F)$. R: $P(E|F)+P(E^{\prime }|F)=P(S|F)=1$.
5. A: X = 0, 1, 2 with probabilities 0.3, 0.5, 0.3 is a probability distribution. R: All its probabilities are non-negative.
6. A: In Bayes' theorem the events $E_i$ must be mutually exclusive and exhaustive. R: The total probability $P(A)=\sum P(E_i)P(A|E_i)$ needs a partition.

Key: 1 (a) · 2 (a) · 3 (d) · 4 (a) · 5 (d) · 6 (a)

Answers to the practice set