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Probability MCQ for Bank Exams

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DIRECTIONS (Q. 11-14) : Study the given information carefully to answer the questions that follow.

An urn contains 4 green, 5 blue, 2 red and 3 yellow marbles.

11. If four marbles are drawn at random, what is the probability that two are blue and two are red?

(a) \(\displaystyle \frac{{10}}{{1001}}\)

(b) \(\displaystyle \frac{{9}}{{14}}\)

(c) \(\displaystyle \frac{{17}}{{364}}\)

(d) \(\displaystyle \frac{{2}}{{7}}\)

(e) None of these

Solution: (a)

According to question,

\(\displaystyle n(S){{=}^{{14}}}{{C}_{4}}=\frac{{14!}}{{(14-4)!4!}}=\frac{{14!}}{{10!4!}}\)

\(\displaystyle \frac{{14\times 13\times 12\times 11}}{{4\times 3\times 2\times 1}}=1001 [As, {{n}}{{C}_{r}}=\frac{{n!}}{{(n-r)!n!}}]\)

\(\displaystyle n(S) \)= \(\displaystyle ^{5}{{C}_{2}}{{\times }^{2}}{{C}_{2}}=\frac{{5!}}{{(5-2)!2!}}\times \frac{{2!}}{{(2-2)!2!}}\)

=\(\displaystyle \frac{{5\times 4}}{{2\times 1}}\times \frac{{2\times 1}}{{1\times 2\times 1}}=10\)

Therefore, Required probability = \(\displaystyle \frac{{n(E)}}{{n(S)}}=\frac{{10}}{{1001}}\)

12. If eight marbles are drawn at random, what is the probability that there are equal number of marbles of each colour ?

(a) \(\displaystyle \frac{4}{7}\)

(b) \(\displaystyle \frac{361}{728}\)

(c) \(\displaystyle \frac{60}{1001}\)

(d) \(\displaystyle \frac{1}{1}\)

(e) None of these

Solution: (c)

According to question,

n(S)= \(\displaystyle ^{{14}}{{C}_{8}}=\frac{{14!}}{{(14-8)!8!}}\times \frac{{14!}}{{6!8!}}\)

\(\displaystyle \frac{{14\times 13\times 12\times 11\times 10\times 9}}{{6\times 5\times 4\times 3\times 2\times 1}}=3003\)

and n(E) = \(\displaystyle ^{4}{{C}_{2}}{{\times }^{5}}{{C}_{2}}{{\times }^{2}}{{C}_{2}}{{\times }^{3}}{{C}_{2}}\)

=\(\displaystyle \frac{{4!}}{{(4-2)!2!}}\times \frac{{5!}}{{(5-2)!2!}}\times \frac{{2!}}{{(2-2)!2!}}\times \frac{{3!}}{{(3-2)!2!}}\)

=\(\displaystyle \frac{{4!}}{{2!2!}}\times \frac{{5!}}{{3!2!}}\times \frac{{2!}}{{0!2!}}\times \frac{{3!}}{{1!2!}}\)

=\(\displaystyle \frac{{4\times 3}}{{2\times 1}}\times \frac{{5\times 4}}{{2\times 1}}\times \frac{1}{1}\times \frac{3}{1}=180\)

Therefore, Required probability = \(\displaystyle \frac{{n(S)}}{{n(E)}}=\frac{{180}}{{3003}}=\frac{{60}}{{1001}}\)

13. If two marbles are drawn at random, what is the probability that both are red or at least one is red ?

(a) \(\displaystyle \frac{{26}}{{91}}\)

(b) \(\displaystyle \frac{{1}}{{7}}\)

(c) \(\displaystyle \frac{{199}}{{364}}\)

(d) \(\displaystyle \frac{{133}}{{191}}\)

(e) None of these

Solution: (e)

n(S) = \(\displaystyle ^{{14}}{{C}_{2}}=\frac{{14!}}{{(14-2)!2!}}=\frac{{14\times 13}}{{2\times 1}}=91\)

Therefore, Probability of at least one red ball                

= \(\displaystyle 1-\frac{{^{{12}}{{C}_{2}}}}{{^{{14}}{{C}_{2}}}}=1-\frac{{66}}{{91}}=\frac{{91-66}}{{91}}=\frac{{25}}{{91}}\)

14. If three marbles are drawn at random, what is the probability that at least one is yellow ?

(a) \(\displaystyle \frac{{1}}{{3}}\)

(b) \(\displaystyle \frac{{199}}{{364}}\)

(c) \(\displaystyle \frac{{165}}{{364}}\)

(d) \(\displaystyle \frac{{3}}{{11}}\)

(e) None of these

Solution: (b)

According to question,

 n(S)=\(\displaystyle ^{{14}}{{C}_{3}}=\frac{{14!}}{{(14-3)!3!}}=\frac{{14\times 13\times 12}}{{3\times 2\times 1}}=364\)

Required probability = \(\displaystyle 1-\frac{{^{{11}}{{C}_{3}}}}{{^{{14}}{{C}_{3}}}}=1-\frac{{165}}{{364}}=\frac{{364-165}}{{364}}=\frac{{199}}{{364}}\)

15. Out of five girls and three boys, four children are to be randomly selected for a quiz contest. What is the probability that all the selected children are girls?

(a) \(\displaystyle \frac{1}{{14}}\)

(b) \(\displaystyle \frac{1}{{7}}\)

(c) \(\displaystyle \frac{5}{{17}}\)

(d) \(\displaystyle \frac{2}{{17}}\)

(e) None of these

Solution: (a)

Total number of ways of selecting 4 children out of 8

\(\displaystyle ^{8}{{C}_{4}}=\frac{{8\times 7\times 6\times 5}}{{1\times 2\times 3\times 4}}=70\)

Number of ways of selecting 4 girls out of 5 = \(\displaystyle ^{5}{{C}_{4}}=5\)

Required probability = \(\displaystyle \frac{5}{{70}}=\frac{1}{{14}}\)

We hope the Probability questions with answers you have attempted here were helpful in your preparation. MCQ on Probability is beneficial for competitive exams like Bank PO, Bank Clerk, SBI PO, SBI Clerk, IBPS PO, IBPS Clerk, IBPS RRB, SSC, SSC CGL, SSC CHSL, SSC GD, Railways, RRB NTPC, RRB Group D, State PSCs, Police, CAPF, and Defence exams like NDA, CDS, and AFCAT.

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