GATE CSE
First time here? Checkout the FAQ!
x
+8 votes
303 views

Three dice are rolled independently. What is the probability that the highest and the lowest value differ by 4?  

  1. $\frac{1}{3}$  
  2. $\frac{1}{6}$  
  3. $\frac{1}{9}$  
  4. $\frac{5}{18}$  
  5. $\frac{2}{9}$
asked in Probability by Veteran (319k points) 575 1442 2957 | 303 views
2/9 is correct.

6 Answers

+2 votes
Best answer
Case 1:largest is 5 , smallest 1 and middle is 2 or 3 or 4 : 3*3!

Case 2.largest is 5 , smallest 1 and middle is 1 or 5 : 3!*2/2!

Case 3:largest is 6 , smallest 2 and middle is 3 or 4 or 5 : 3*3!

Case 4:largest is 6 , smallest 2 and middle is 6 or 2: 3!*2/2!

So probability the highest and the lowest value differ by 4 =( 3*3!+3!*2/2!+3*3!+3!*2/2!)/6^3 =2/9
answered by Loyal (4.3k points) 4 24 57
selected by
+4 votes

suppose three dices are there A,B,C, three are rolled independently, see we get 4 difference between max and min when we get either (1,5) or (2,6) in any of the two dices. okay?

1st case: we get (1,5), 1 and 5 can be the result of any of the two dice in 3C2=3 ways, either (A-B) or (B-C) and (C-A) and also they can be permutated between them,so 3C2*2! ways possible, now for each way 3rd result can come either 1 or 2 or 3 or 4 or 5(6 cant be possible as min:1 and max:5) But we cant do 3C2*2!*5 because then (1,1,5), (1,5,1),(5,1,1),(1,5,5),(5,1,5),(5,5,1) all these 6 ways are counted two times.So we do like below:
When 3rd result is 1, no of ways=3!/2!=3  because its no of permutation of 1,1 and 5

When 3rd result is 5,no of ways=3!/2!=3 because its no of permutation of 1,5 and 5

when 3rd result is 2 or 3 or 4, 3C2*2!*3=18 ways possible.so totally 18+3+3=24

2nd case: we get (2,6), 2 and 6 can be the result of any of the two dice in 3C2=3 ways, either (A-B) or (B-C) and (C-A) and also they can permutate between them,so 3C2*2! ways possible, now for each way 3rd dice result can come either 2 or 3 or 4 or 5 or 6(min:2 and max:6) But we cant do 3C2*2!*5, So we do like below:

Similarly as shown above, when 3rd result is 2 or 6,no of ways=2*3!/2=6 ways possible,when 3rd result is 3 or 4 or 5,3C2*2!*3 =18 ways possible.So totally 18+6=24 ways possible.


Finally total=24+24=48 ways possible
sample space= 6^3=216
so probability= 48/216=2/9

answered by Junior (517 points) 1 3 10
edited by
+1 vote
difference highest and lowest is 4.
1 (1,2,3,4,5) 5
+
2(2,3,4,5,6)6
=(3!* 1.5.1 + 3!*1.5.1)/6.6.6
=2.6.5/6.6.6
=10/36=5/18
answered by Loyal (3.1k points) 5 36 78
+1 vote
Sno Lowest Value Highest Value  Values at dice No of ways
1 1 5 {1,1,5} 3
2 1 5 {1,2,5} 6
3 1 5 {1,3,5} 6
4 1 5 {1,4,5} 6
5 1 5 {1,5,5} 3
6 2 6 {2,2,6} 3
7 2 6 {2,3,6} 6
8 2 6 {2,4,6} 6
9 2 6 {2,5,6} 6
10 2 6 {2,6,6} 3

 

By naive defn of Prob = # outcomes which satisfy our constraint / total # of outcomes

Prob = 48 / 63 = 2 / 9 

The correct answer is (E) 2/9

 

answered by Boss (8.8k points) 3 8 12
0 votes
1/9 C is answer .  there will be 24 such permutations having diff. of 4 in max. and min. in dies

24/216=1/9
answered by (25 points) 1 2
0 votes

There will be 2 possibilities for highest and lowest differ by 4, i.e (1,5),(2,6) and they can arrange in 2! ways

Now among 3 dies 2 dice can be selected in 3C2 ways

Case 1:

for (1,5) the 3rd dice value can be any value from 1 to 5, but it cannot be 6, Because in that case difference cannot be 4

so probability is 3C2*2!*5/ (6^3)

Case 2:

for (2,6) the 3rd dice value can be any value from 2 to 6, but it cannot be 1, Because in that case difference cannot be 4

so probability is 3C2*2!*5/ (6^3)

So,  probability that the highest and the lowest value differ by 4

=3C2*2!*5/ (6^3)+3C2*2!*5/ (6^3)= 5/18

Ans is (D)

 

 

answered by Veteran (64.6k points) 35 220 622


Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
Top Users Oct 2017
  1. Arjun

    23210 Points

  2. Bikram

    17018 Points

  3. Habibkhan

    6652 Points

  4. srestha

    5864 Points

  5. Debashish Deka

    5430 Points

  6. jothee

    4908 Points

  7. Sachin Mittal 1

    4762 Points

  8. joshi_nitish

    4274 Points

  9. sushmita

    3954 Points

  10. Silpa

    3698 Points


Recent Badges

Regular Juhi Sehgal
Popular Question vineet.ildm
Nice Comment Arjun
100 Club vipul verma
Notable Question jothee
Popular Question jothee
Nice Question shivangi5
Regular rinks5
Notable Question shipra tressa
Regular sasi
27,247 questions
35,056 answers
83,703 comments
33,183 users