Probability Questions and Answers

Q: A dice is rolled twice. What is the probability of getting sum 9?

Two rolls give 36 outcomes. Those totalling 9 are\[ (3,6),\ (4,5),\ (5,4),\ (6,3) \]\[ P = \frac{4}{36} = \frac{1}{9} \]The order of the two rolls matters, so \( (3,6) \) and \( (6,3) \) are separate outcomes; treating them as one would halve the count.

Q: A box has 10 black and 10 white balls. What is the probability of getting two balls of the same color?

There are 20 balls in all.\[ \text{total ways} = {}^{20}C_{2} = 190 \]\[ \text{same colour} = {}^{10}C_{2} + {}^{10}C_{2} = 45 + 45 = 90 \]\[ P = \frac{90}{190} = \frac{9}{19} \]The complement - one of each colour - has probability \( \tfrac{100}{190} = \tfrac{10}{19} \), and the two add to 1.

Q: The sum of the probabilities of all elementary events of an experiment is p, then

The elementary events of an experiment are mutually exclusive and between them cover every possible outcome, so their probabilities must account for the whole.\[ p = 1 \]This is the normalisation axiom of probability. Any set of assignments summing to less or more than 1 is not a valid probability distribution.

Important Instructions

81.

Two dice are tossed simultaneously. Find the probability that the total is a prime number.

A.7/9
B.5/12
C.1/6
D.5/9

Answer: B

Two dice give 36 outcomes. The prime totals are 2, 3, 5, 7 and 11.

sum 2: 1 way
sum 3: 2 ways
sum 5: 4 ways
sum 7: 6 ways
sum 11: 2 ways

\[ 1 + 2 + 4 + 6 + 2 = 15 \quad\Longrightarrow\quad P = \frac{15}{36} = \frac{5}{12} \]

9 is not prime (\( 3 \times 3 \)) and neither is 1, which is where miscounts usually creep in.

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82.

A class consists of 15 girls and 10 boys. Three students are to be randomly selected. Find the probability that one boy and two girls are picked.

A.1/50
B.3/25
C.21/46
D.25/122

Answer: C

Choose one boy from 10 and two girls from 15, out of all ways to choose 3 from 25.

\[ P = \frac{{}^{10}C_{1} \times {}^{15}C_{2}}{{}^{25}C_{3}} = \frac{10 \times 105}{2300} \]

\[ = \frac{1050}{2300} = \frac{21}{46} \]

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83.

A dice is rolled twice. What is the probability of getting sum 9?

A.2/3
B.1/3
C.1/9
D.3/9

Answer: C

Two rolls give 36 outcomes. Those totalling 9 are

\[ (3,6),\ (4,5),\ (5,4),\ (6,3) \]

\[ P = \frac{4}{36} = \frac{1}{9} \]

The order of the two rolls matters, so \( (3,6) \) and \( (6,3) \) are separate outcomes; treating them as one would halve the count.

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84.

A box has 10 black and 10 white balls. What is the probability of getting two balls of the same color?

A.10/19
B.9/38
C.9/19
D.5/38

Answer: C

There are 20 balls in all.

\[ \text{total ways} = {}^{20}C_{2} = 190 \]

\[ \text{same colour} = {}^{10}C_{2} + {}^{10}C_{2} = 45 + 45 = 90 \]

\[ P = \frac{90}{190} = \frac{9}{19} \]

The complement - one of each colour - has probability \( \tfrac{100}{190} = \tfrac{10}{19} \), and the two add to 1.

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85.

Three coins are tossed. What is the probability of getting at most two tails?

A.7/8
B.1/8
C.2/8
D.4/8

Answer: A

It is given that 3 coins are tossed simultaneously.
=>Total outcomes when one coin is tossed=2
=>Total outcomes when 3 coins are tossed= 2n = 2 X 2 X 2
=8 outcomes
Favourable outcomes: TTH, THT, HTT, THH, THT,HHT,HHH=7
Therefore, Probability= 7/8

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86.

The probability of getting exactly one head in tossing a pair of coins is

A.1
B.1/3
C.1/2

Answer: C

Two coins give four outcomes: HH, HT, TH, TT. Exactly one head occurs in HT and TH.

\[ P = \frac{2}{4} = \frac{1}{2} \]

HT and TH are distinct outcomes because the coins are distinguishable; treating them as one would give \( \tfrac{1}{3} \).

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87.

The total number of events of throwing 10 coins simultaneously is

A.1024
B.512
C.100
D.10

Answer: A

Each coin can land in 2 ways, and the ten coins are independent.

\[ 2^{10} = 1024 \]

The rule is \( 2^{n} \) for \( n \) coins - not \( 2n \), which would give only 20.

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88.

The total events to throw three dice simultaneously is

A.6
B.18
C.81
D.216

Answer: D

Each die shows one of 6 faces, and the three dice are independent.

\[ 6^{3} = 216 \]

The rule is \( 6^{n} \) for \( n \) dice; \( 3^{4} = 81 \) and \( 3 \times 6 = 18 \) are the shapes of the two nearest distractors.

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89.

A fair dice is rolled. Probability of getting a number x such that 1 ≤ x ≤ 6, is

A.> 1
B.between 0 and 1
C.1

Answer: C

Every face of a die shows a number \( x \) with \( 1 \leq x \leq 6 \), so the event covers all six outcomes.

\[ P = \frac{6}{6} = 1 \]

An event that is bound to happen is a sure event, with probability exactly 1. No probability can exceed 1.

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90.

The sum of the probabilities of all elementary events of an experiment is p, then

A.0 < p < 1
B.0 ≤ p < 1
C.p = 1
D.p = 0

Answer: C

The elementary events of an experiment are mutually exclusive and between them cover every possible outcome, so their probabilities must account for the whole.

\[ p = 1 \]

This is the normalisation axiom of probability. Any set of assignments summing to less or more than 1 is not a valid probability distribution.

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