Question

Five marbles are drawn from a bag which contains 7 blue marbles and 4 blackk marbles. Find out the probability of (i) all will be blue (ii) 3 will be blue and 2 black​

Answer 1

The probability that all will be blue is 1/22

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Answer 2

Step-by-step explanation:

$\bf{\underline{\underline\red{Given:-}}}$

• Five balls are drawn from the bag

• The bag contains

• 7 blue marbles

• 4 black marbles

$\bf{\underline{\underline\blue{To\:Find:-}}}$

• The probability of all the marbles drawn are blue.

• The probability of 3 marbles will be blue and 2 black.

$\bf{\underline{\underline\green{Solution:-}}}$

Given

The bag contains 7 blue marbles and 4 black marbles.

The total number of marbles

= 7 + 4

= 11

Number of possibilities of taking 5 marbles out of 11 marbles = $\rm ^{n} C_{r} = \: ^{11} C_{5}$

Number of possibilities of taking 5 marbles out of 7 marbles = $\rm = \: ^{7} C_{5}$

The probability that all the balls drawn are blue

$\implies \rm \dfrac{^{7} C_{5} }{ ^{11}C_{5}}$

As we know that:-

$\boxed{ \leadsto\rm ^{n} C_{r} = \frac{n! }{r!(n - r)!} }$

$\implies \rm \dfrac{^{7} C_{5} }{ ^{11}c_{5}}$

$\implies \rm \dfrac{ \dfrac{7!}{5!(7 - 5) ! } }{ \dfrac{11! }{5!(11 - 5)!} }$

$\implies \rm \dfrac{ \dfrac{7!}{5! \: 2! } }{ \dfrac{11! }{5! \: 6!} }$

$\implies \rm \dfrac{7!}{5! \: 2! } \times { \dfrac{5! \: 6!}{11!}}$

$\implies \rm \dfrac{ \not7!}{ \not5! \: 2! } \times { \dfrac{ \not5! \: 6!}{11 \times 10 \times 9 \times 8 \times \not7!}}$

$\implies \rm \dfrac{1}{22}$

(i) Probability that all the marbles drawn are blue = 1/22

Now, probability that the marbles drawn are 3 blue and 2 black.

Number of possibilities of taking 3 blue marbles from 7

= $\rm \: ^{7} C_{3}$

Number of possibilities of taking 3 blue marbles from 4

= $\rm \: ^{4} C_{2}$

Total number of possibilities = $\rm \: ^{11} C_{5}$

Probability that the marbles drawn are 3 blue and 2 black.

$\implies \rm \dfrac{^{7} C_{3} + ^{4}C_{2}}{ ^{11}C_{5}}$

$\implies \rm \dfrac{35+ 6}{ 462}$

$\implies \rm \dfrac{41}{ 462}$

(ii) Probability that all the marbles 3 blue and 3 black = 41/462