A set of billiard balls consists of 16 balls: 1 cue ball, 15 balls numbered 1 to

Question

A set of billiard balls consists of 16 balls: 1 cue ball, 15 balls numbered 1 to 15 respectively. Batts from 1 to 8 are labeled solids. Those from 9 to 15 are labeled stripes. Three sets of these have been placed in a box. Balls are drawn without replacement. (Questions are independent of each other) a) If three balls are drawn from the box, what is the probability that all three balls are Identical? b) If four balls are drawn from the box, what is the probability that three of them are stripes? c) If two balls are drawn, what Is the probability that their sum is 28? d) Stripes balls have been taken out of the box. If seven balls are drawn, what is the probability that two of them are 3s, three of them 2s, and two of them are cue balls? e) All odd-numbered balls are taken out of the box. If four balls are drawn, what is the probability that at least three of them are divisible by four?

Explanation / Answer

number of balls = 16

in that 1 cue ball

1 to 8 are solid balls

9 to 15 are labeled stripes

a) three balls are drawn probability that they are identical

number of ways total sum is 16c3= 16*15*14 / 3*2*1 = 560

Total cases of identical are 8c3 +9c3 =8*7*6 / 3*2*1 + 9*8*7 / 3*2.1 = 56+84 =140

probability is = 140/560= 14/56 = 1/4 = 0.25

b) four balls drawn from box whart is the probability three of them are stripes

number of stripe balls are 9 to 15 so total 7

three of them are stripes one ball will be solids so

prob is 7 C3 +8 C1 = 7*6*5 / 3*2*1 = 35 +8 = 43

TOT PROB = 43/540= 0.08

C) Two balls are drawn probility that there sum is 28

balls which gives us sum 28 is

Out of 8C2=8C2=28 possible sums,

only 4 (7 + 0, 6 + 1, 5 + 2, 4 + 3)

probaility = 4/28=1/7 =0.142

d)seven balls are drawn

7c3+7c2+7c2 = 7*6*5 / 3*2*1 + 7*6 / 2*1 + 7*6/2*1

= 35+21+21 =77

probbaility is 77/540=0.142

e) ood number of balls are taken

four balls are drawn

1 , 3, 5, 7, 9, 11 , 13 , 15 , =8 odd balls taken out means even balls there in that four balls are taken so

2, 4, 6, 8, 10 ,1 2 , 14 =7 even balls

divisible by 4 = (4,6,12) =3

probability is 3/7 = 0.428

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