Probability

1

MCQ · Numerical

If P(A) = 7/13, P(B) = 9/13 and P(A ∩ B) = 4/13, evaluate P(A|B).

A

4/7

B

4/9

C

7/9

D

9/13

✔ Solution

P(A|B) = P(A∩B)/P(B) = (4/13)/(9/13) = 4/9.

Answer: B — 4/9

2

MCQ · Numerical

A family has two children. What is the probability that both are boys, given that at least one is a boy?

A

1/4

B

1/2

C

1/3

D

2/3

✔ Solution

S = {bb,bg,gb,gg}. E = {bb}, F = {bb,bg,gb}. P(F)=3/4, P(E∩F)=1/4.

P(E|F) = (1/4)/(3/4) = 1/3.

Answer: C — 1/3

3

MCQ · Numerical

Cards numbered 1–10 are in a box. One is drawn. Given the number is more than 3, what is the probability it is even?

A

1/2

B

4/7

C

3/7

D

2/5

✔ Solution

A={2,4,6,8,10}, B={4,5,6,7,8,9,10}, A∩B={4,6,8,10}.

P(A|B) = (4/10)/(7/10) = 4/7.

Answer: B — 4/7

4

MCQ · Numerical

A die is thrown 3 times. A: 4 on the third throw; B: 6 and 5 on first two throws. Find P(A|B).

A

1/36

B

1/6

C

1/216

D

1/3

✔ Solution

P(B)=6/216, P(A∩B)=1/216 (only (6,5,4)).

P(A|B)=(1/216)/(6/216)=1/6.

Answer: B — 1/6

5

MCQ · Concept

If P(A)=1/2, P(B)=0, then P(A|B) is:

A

0

B

1/2

C

Not defined

D

1

✔ Solution

P(A|B)=P(A∩B)/P(B). Since P(B)=0, division by zero → not defined.

Answer: C — Not defined

6

MCQ · Numerical

An urn has 10 black and 5 white balls. Two balls are drawn without replacement. Probability that both are black?

A

2/7

B

3/7

C

4/7

D

1/3

✔ Solution

P(E)=10/15=2/3. P(F|E)=9/14 (9 black left out of 14).

P(E∩F)=(10/15)×(9/14)=90/210=3/7.

Answer: B — 3/7

7

MCQ · Numerical

Three cards drawn successively without replacement from 52. Probability first two are kings and third is ace?

A

2/5525

B

1/5525

C

4/5525

D

3/5525

✔ Solution

P(K)=4/52, P(K|K)=3/51, P(A|KK)=4/50.

P(KKA)=(4×3×4)/(52×51×50)=48/132600=2/5525.

Answer: A — 2/5525

8

MCQ · Numerical

A die thrown. E={3,6}, F={2,4,6}. Find P(E∩F) and check independence.

A

P(E∩F)=1/6, independent

B

P(E∩F)=1/3, dependent

C

P(E∩F)=1/6, dependent

D

P(E∩F)=1/2, independent

✔ Solution

E∩F={6}. P(E∩F)=1/6. P(E)·P(F)=(1/3)×(1/2)=1/6. Since equal, events are independent.

Answer: A

9

MCQ · Numerical

Bag I: 3 red, 4 black. Bag II: 5 red, 6 black. A random bag is chosen, a red ball drawn. P(from Bag II)?

A

35/68

B

33/68

C

5/11

D

3/7

✔ Solution

P(E₁)=P(E₂)=1/2. P(A|E₁)=3/7, P(A|E₂)=5/11. By Bayes:

P(E₂|A)=[(1/2)(5/11)]/[(1/2)(3/7)+(1/2)(5/11)]=35/68.

Answer: A — 35/68

10

MCQ · Numerical

P(strike)=0.65, P(complete|no strike)=0.80, P(complete|strike)=0.32. Find P(job completed on time) using Total Probability Theorem.

A

0.56

B

0.488

C

0.42

D

0.32

✔ Solution

P(A)=P(B)·P(A|B)+P(B')·P(A|B')=0.65×0.32+0.35×0.80=0.208+0.280=0.488.

Answer: B — 0.488

11

MCQ · Numerical

A man speaks truth 3 out of 4 times. He reports a die shows 6. What is the probability it actually is a six?

A

1/2

B

3/8

C

3/5

D

1/4

✔ Solution

P(S₁)=1/6, P(S₂)=5/6, P(E|S₁)=3/4, P(E|S₂)=1/4.

P(S₁|E)=[(1/6)(3/4)]/[(1/6)(3/4)+(5/6)(1/4)]=(3/24)/(8/24)=3/8.

Answer: B — 3/8

12

MCQ · Numerical

Machines A, B, C produce 25%, 35%, 40% of bolts. Defect rates: 5%, 4%, 2%. A defective bolt is found. P(from B)?

A

28/69

B

25/69

C

14/69

D

16/69

✔ Solution

Denominator=0.25×0.05+0.35×0.04+0.40×0.02=0.0125+0.0140+0.0080=0.0345.

P(B₂|E)=0.0140/0.0345=28/69.

Answer: A — 28/69

13

MCQ · Concept

If A and B are events such that P(A|B) = P(B|A), then which is true?

A

A ⊂ B but A ≠ B

B

A = B

C

A ∩ B = ϕ

D

P(A) = P(B)

✔ Solution

Setting P(A∩B)/P(B) = P(A∩B)/P(A) gives P(A) = P(B).

Answer: D — P(A) = P(B)

14

MCQ · Numerical

An unbiased die is thrown twice. A: odd on first, B: odd on second. Are A and B independent? Find P(A∩B).

A

1/2, dependent

B

1/4, independent

C

1/6, independent

D

1/4, dependent

✔ Solution

P(A)=P(B)=1/2. P(A∩B)=9/36=1/4 = P(A)·P(B) → independent, P(A∩B)=1/4.

Answer: B — 1/4, independent

15

MCQ · Numerical

Probability of getting an even prime number on each die when a pair of dice is rolled:

A

0

B

1/3

C

1/12

D

1/36

✔ Solution

Only even prime = 2. P(2 on each die) = 1/6 × 1/6 = 1/36.

Answer: D — 1/36

16

One-Word Ans. · Numerical

If P(E)=0.6, P(F)=0.3, P(E∩F)=0.2, find P(E|F).

✔ Solution

P(E|F)=P(E∩F)/P(F)=0.2/0.3=2/3.

Answer: 2/3

17

One-Word Ans. · Numerical

If P(A)=3/5, P(B)=1/5 and A, B are independent events, find P(A∩B).

✔ Solution

P(A∩B)=P(A)×P(B)=(3/5)×(1/5)=3/25.

Answer: 3/25

18

One-Word Ans. · Numerical

Three coins are tossed. Event E = 'at least two heads'. Find P(E).

✔ Solution

E = {HHH, HHT, HTH, THH} → 4 outcomes out of 8. P(E)=4/8=1/2.

Answer: 1/2

19

One-Word Ans. · Numerical

A die is thrown twice. The sum is 6. How many sample points satisfy this event?

✔ Solution

Pairs: (1,5),(2,4),(3,3),(4,2),(5,1) → 5 outcomes.

Answer: 5

20

One-Word Ans. · Numerical

If P(A)=0.8, P(B)=0.5, P(B|A)=0.4, find P(A∩B).

✔ Solution

By multiplication rule: P(A∩B)=P(A)·P(B|A)=0.8×0.4=0.32.

Answer: 0.32

21

One-Word Ans. · Numerical

A die is thrown three times. How many total outcomes are in the sample space?

✔ Solution

Each throw: 6 outcomes. Total = 6³ = 216.

Answer: 216

22

One-Word Ans. · Numerical

A and B throw a die alternately (A starts). A wins by getting 6. Find P(B wins).

✔ Solution

P(A wins)=6/11. P(B wins)=1−6/11=5/11.

Answer: 5/11

23

One-Word Ans. · Numerical

A coin is tossed. If head appears, it is tossed again; if tail, a die is thrown. How many outcomes total in sample space?

✔ Solution

Head → {HH, HT}: 2 outcomes. Tail → {T,1},{T,2},{T,3},{T,4},{T,5},{T,6}: 6 outcomes. Total = 8.

Answer: 8

24

One-Word Ans. · Numerical

Three boxes I, II, III each contain 2 coins. Box I: both gold; Box II: both silver; Box III: one gold, one silver. A gold coin is drawn. P(other coin is also gold) =?

✔ Solution

P(E₁)=P(E₂)=P(E₃)=1/3. P(A|E₁)=1, P(A|E₂)=0, P(A|E₃)=1/2.

P(E₁|A)=(1/3×1)/[(1/3×1)+(1/3×0)+(1/3×1/2)]=(1/3)/(1/3+1/6)=2/3.

Answer: 2/3

25

One-Word Ans. · Numerical

P(E)=3/5, P(F)=3/10, P(E∩F)=1/5. Are E and F independent? (Answer: Yes or No)

✔ Solution

P(E)·P(F) = (3/5)×(3/10) = 9/50. But P(E∩F) = 1/5 = 10/50. Since 9/50 ≠ 10/50, events are NOT independent.

Answer: No (Not independent)

Multiple Choice Questions

One-Word / One-Value Answer (Numerical)

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