Probability - Aptitude MCQ Questions and Solutions with Explanations

11

Out of 10 persons working on a project, 4 are graduates. If 3 are selected, what is the probability that there is at least one graduate among them?

A) $$\frac{{1}}{{6}}$$ B) $$\frac{{5}}{{8}}$$ C) $$\frac{{3}}{{8}}$$ D) $$\frac{{5}}{{6}}$$

Discuss

Answer & Solution

Answer: Option D

Solution:

P(at least one graduate) = 1 - P(no graduates)
$$\eqalign{ & = 1 - \frac{{{}^6{C_3}}}{{{}^{10}{C_3}}} \cr & = 1 - \frac{{6 \times 5 \times 4}}{{10 \times 9 \times 8}} \cr & = \frac{5}{6} \cr} $$

12

A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If four marbles are picked at random, what is the probability that none is blue?

A) $$\frac{{17}}{{91}}$$ B) $$\frac{{33}}{{91}}$$ C) $$\frac{{51}}{{91}}$$ D) $$\frac{{65}}{{91}}$$

Discuss

Answer & Solution

Answer: Option B

Solution:

Given that there are three blue marbles, four red marbles, six green marbles and two yellow marbles.
When four marbles are picked at random, then the probability that none is blue is
$$\eqalign{ & = \frac{{{}^{12}{C_4}}}{{{}^{15}{C_4}}} \cr & = \frac{{12 \times 11 \times 10 \times 9}}{{15 \times 14 \times 13 \times 12}} \cr & = \frac{{11880}}{{32760}} \cr & = \frac{{33}}{{91}} \cr} $$

13

The probability that a number selected at random from the first 50 natural numbers is a composite number is -

A) $$\frac{{21}}{{25}}$$ B) $$\frac{{17}}{{25}}$$ C) $$\frac{{4}}{{25}}$$ D) $$\frac{{8}}{{25}}$$

Discuss

Answer & Solution

Answer: Option B

Solution:

The number of exhaustive events = $${{}^{50}{C_1}}$$ = 50
We have 15 primes from 1 to 50
Number of favourable cases are 34
Required probability = $$\frac{{34}}{{50}}$$ = $$\frac{{17}}{{25}}$$

14

If a card is drawn from a well shuffled pack of cards, the probability of drawing a spade or a king is -

A) $$\frac{{19}}{{52}}$$ B) $$\frac{{17}}{{52}}$$ C) $$\frac{{5}}{{13}}$$ D) $$\frac{{4}}{{13}}$$

Discuss

Answer & Solution

Answer: Option D

Solution:

$$P\left( {S \cup K} \right) = $$ $$P\left( S \right) + $$ $$P\left( K \right) - $$ $$P\left( {S \cap K} \right),$$ (where S denotes spade and K denotes king.)
$$\eqalign{ & P\left( {S \cup K} \right) = \frac{{13}}{{52}} + \frac{4}{{52}} - \frac{1}{{52}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{4}{{13}} \cr} $$

15

A bag contains five white and four red balls. Two balls are picked at random from the bag. What is the probability that they both are different color?

A) $$\frac{{4}}{{9}}$$ B) $$\frac{{5}}{{9}}$$ C) $$\frac{{7}}{{9}}$$ D) $$\frac{{8}}{{9}}$$

Discuss

Answer & Solution

Answer: Option B

Solution:

Two balls can be picked from nine balls in $${{}^9{C_2}}$$ ways.
We select one white ball and one red ball from five white balls and four red balls.
This can be done $${{}^5{C_1}}$$ . $${{}^4{C_1}}$$ ways.
∴ The required probability
$$\eqalign{ & = \frac{{5 \times 4}}{{{}^9{C_2}}} \cr & = \frac{{20}}{{36}} \cr & = \frac{5}{9} \cr} $$

16

A box contains nine bulbs out of which 4 are defective. If four bulbs are chosen at random, find the probability that exactly three bulbs are good.

A) $$\frac{{20}}{{31}}$$ B) $$\frac{{20}}{{63}}$$ C) $$\frac{{5}}{{31}}$$ D) $$\frac{{6}}{{31}}$$

Discuss

Answer & Solution

Answer: Option B

Solution:

Required probability
$$\eqalign{ & = \frac{{{}^5{C_3}\,.\,{}^4{C_1}}}{{{}^9{C_4}}} \cr & = \frac{{10 \times 4}}{{126}} \cr & = \frac{{20}}{{63}} \cr} $$

17

A basket has 5 apples and 4 oranges. Three fruits are picked at random. The probability that at least 2 apples are picked is -

A) $$\frac{{25}}{{42}}$$ B) $$\frac{{9}}{{20}}$$ C) $$\frac{{10}}{{23}}$$ D) $$\frac{{41}}{{42}}$$

Discuss

Answer & Solution

Answer: Option A

Solution:

Total fruits = 9
Since there must be at least two apples,
$$\eqalign{ & = \frac{{{}^5{C_2}\, \times \,{}^4{C_1}}}{{{}^9{C_3}}} + \frac{{{}^5{C_3}}}{{{}^9{C_3}}} \cr & = \frac{{25}}{{42}} \cr} $$

18

Out of first 20 natural numbers, one number is selected at random. The probability that it is either an even number or a prime number is -

A) $$\frac{{1}}{{2}}$$ B) $$\frac{{16}}{{19}}$$ C) $$\frac{{4}}{{5}}$$ D) $$\frac{{17}}{{20}}$$

Discuss

Answer & Solution

Answer: Option D

Solution:

$$\eqalign{ & n\left( S \right) = 20 \cr & n\left( {{\text{Even no}}} \right) = 10 = n\left( E \right) \cr & n\left( {{\text{Prime no}}} \right) = 8 = n\left( P \right) \cr & P\left( {E \cup P} \right) = \frac{{10}}{{20}} + \frac{8}{{20}} - \frac{1}{{20}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \frac{{17}}{{20}} \cr} $$

19

The probability that A speaks truth is $$\frac{{3}}{{5}}$$ and that of B speaking truth is $$\frac{{4}}{{7}}$$. What is the probability that they agree in stating the same fact?

A) $$\frac{{18}}{{35}}$$ B) $$\frac{{12}}{{35}}$$ C) $$\frac{{17}}{{35}}$$ D) $$\frac{{19}}{{35}}$$

Discuss

Answer & Solution

Answer: Option A

Solution:

If both agree stating the same fact, either both of them speak truth of both speak false.
∴ Probability
$$\eqalign{ & = \frac{3}{5} \times \frac{4}{7} + \frac{2}{5} \times \frac{3}{7} \cr & \, = \frac{{12}}{{35}} + \frac{6}{{35}} \cr & = \frac{{18}}{{35}} \cr} $$

20

A box contains nine bulbs out of which 4 are defective. If four bulbs are chosen at random, find the probability that at least one bulb is good.

A) $$\frac{{6}}{{63}}$$ B) $$\frac{{2}}{{63}}$$ C) $$\frac{{125}}{{126}}$$ D) $$\frac{{1}}{{126}}$$

Discuss

Answer & Solution

Answer: Option C

Solution:

Required probability
$$\eqalign{ & = 1 - \frac{1}{{126}} \cr & = \frac{{125}}{{126}} \cr} $$