ব্যাংক জব নিয়োগ পরীক্ষা গণিত Probability

Probability problem on Balls

Examples 1:
In a Bag contains 8 White balls and 6 Black balls from that bag two balls are taken out at random.Find the Probability that they are of the same color ?
Answer :
Here S be the sample space So,
n(S) = Number of ways of  taken out balls of ( 8 + 6 ) = 14 c ₂ = ( 14 x 13 ) / ( 2 x 1 ) = 182 / 2 = 91.
E = Event of getting both balls of the same color . Then,n( E ) = Number of ways of drawing ( 2 balls out of 8 ) or ( 2 balls out of 6 )
= (8 c ₂ + 6 c ₂ ) = 8 x 7 / 2 x 1 + 6 x 5 / 2 x 1 = 56 / 2 + 30 / 2 = 28 + 15 = 43.

P( E ) = n( E ) / n ( S ) = 43 / 91 .

Example 2:
A bag contains 4 red , 6 yellow balls. 2 balls are drawn randomly . What is the probability that they are of the same color ?
Answer :
p(E) = n( E ) / n ( S ) = ⁴c₂ + ⁶c₂ / ¹⁰c₂
( 4 x 3 / 2 + 6 x 5 / 2 ) / 10 x 9 / 2
= 4 + 10 / 30 = 40 / 30
= 4 / 3 .

Example 3:
A bag contains 4 red , 6 yellow  5 green balls. 3 balls are drawn randomly . What is the probability that the balls drawn contain exactly two green balls ?
Answer :
2 green balls can be selected from 5 green balls and the remaining  one ball select from (15 – 5) = 10 balls in 10c1 ways .
n( E )  5c2 x 10c1= 10 x 10 = 100 .
So P ( E ) = 100 / 220 = 5 / 11.

Example 4:
A bag contain 4 red , 6 yellow  5 green balls . 3 balls are drawn randomly . What is the probability that the balls drawn contain balls of different colors ?
Answer :
Total number of balls = 4 + 6 + 5 = 15 ,
n ( S ) = 12c3 = 12 x 11 x 10 / 3 x 2 = 220 .
In order to 3 colored different balls , one ball from each color ,
n ( E ) =4c1 x 6c1 x 5c1 = 4 x 6 x 5 = 120
p(E) = 120 / 220 = 6 / 11 .

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