Probability - Study Mode
[#146] A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If three marbles are picked at random, what is the probability that they are all blue?
Correct Answer
(A) $$frac{{1}}{{455}}$$
Explanation
Solution: Given that there are three blue marbles, four red marbles, six green marbles and two yellow marbles. Probability that all the three marbles picked at random are blue = $$frac{{{}^3{C_3}}}{{{}^{15}{C_3}}}$$ $$eqalign{
& = frac{{1 imes 3 imes 2 imes 1}}{{15 imes 14 imes 13}} cr
& = frac{6}{{2730}} cr
& = frac{1}{{455}} cr} $$
[#147] In a party there are 5 couples. Out of them 5 people are chosen at random. Find the probability that there are at the least two couples?
Correct Answer
(A) $$frac{{5}}{{21}}$$
Explanation
Solution: Number of ways of (selecting at least two couples among five people selected) = $$left( {{}^5{C_2} imes {}^6{C_1}}
ight)$$ As remaining person can be any one among three couples left. Required probability $$eqalign{
& = frac{{{}^5{C_2} imes {}^6{C_1}}}{{{}^{10}{C_5}}} cr
& = frac{{left( {10 imes 6}
ight)}}{{252}} cr
& = frac{5}{{21}} cr} $$
[#148] If a number is chosen at random from the set {1, 2, 3, ......., 100}, then the probability that the chosen number is a perfect cube is -
Correct Answer
(A) $$frac{{1}}{{25}}$$
Explanation
Solution: We have 1, 8, 27 and 64 as perfect cubes from 1 to 100. Thus, the probability of picking a perfect cube is $$eqalign{
& = frac{4}{{100}} cr
& = frac{1}{{25}} cr} $$
[#149] A box contains 3 blue marbles, 4 red, 6 green marbles and 2 yellow marbles. If three marbles are drawn what is the probability that one is yellow and two are red?
Correct Answer
(C) $$frac{{12}}{{455}}$$
Explanation
Solution: Given that there are three blue marbles, four red marbles, six green marbles and two yellow marbles. When three marbles are drawn, the probability that one is yellow and two are red $$eqalign{
& = frac{{left( {{}^2{C_1}}
ight)left( {{}^4{C_2}}
ight)}}{{{}^{15}{C_3}}} cr
& = frac{{2 imes 4 imes 3 imes 3 imes 2}}{{1 imes 2 imes 15 imes 14 imes 13}} cr
& = frac{{144}}{{5460}} cr
& = frac{{12}}{{455}} cr} $$
[#150] From a pack of cards two cards are drawn one after the other, with replacement. The probability that the first is a red card and the second is a king is -
Correct Answer
(A) $$frac{{1}}{{26}}$$
Explanation
Solution: Let E 1 be the event of drawing a red card. Let E 2 be the event of drawing a king. $$Pleft( {{E_1} cap {E_2}}
ight) = Pleft( {{E_1}}
ight).Pleft( {{E_2}}
ight)$$ (As E 1 and E 2 are independent) $$eqalign{
& = frac{1}{2} imes frac{1}{{13}} cr
& = frac{1}{{26}} cr} $$