Mar 19 08:00 AM PDT  09:00 AM PDT Beat the GMAT with a customized study plan based on your needs! Learn how to create your preparation timeline, what makes a good study plan and which tools you need to use to build the perfect plan. Register today! Mar 20 07:00 AM PDT  09:00 AM PDT Join a FREE 1day workshop and learn how to ace the GMAT while keeping your fulltime job. Limited for the first 99 registrants. Mar 20 09:00 PM EDT  10:00 PM EDT Strategies and techniques for approaching featured GMAT topics. Wednesday, March 20th at 9 PM EDT Mar 23 07:00 AM PDT  09:00 AM PDT Christina scored 760 by having clear (ability) milestones and a trackable plan to achieve the same. Attend this webinar to learn how to build trackable milestones that leverage your strengths to help you get to your target GMAT score.
Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 26 Nov 2014
Posts: 90

If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
Updated on: 18 Feb 2019, 05:56
Question Stats:
62% (01:29) correct 38% (01:12) wrong based on 269 sessions
HideShow timer Statistics
If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled? A. 25/216 B. 50/216 C. 25/72 D. 25/36 E. 5/6
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
Consider Kudos for my post, if it is helpful. TIA
Originally posted by shyind on 04 Nov 2015, 05:15.
Last edited by Bunuel on 18 Feb 2019, 05:56, edited 1 time in total.
Updated.




Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8982
Location: Pune, India

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
04 Nov 2015, 05:39
shyind wrote: If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled?
A. 25/216
B. 50/216
C. 25/72
D. 25/36
E. 5/6 Total ways in which a 6sided die can be rolled three times = 6*6*6 = 216 To get exactly one 3, there are three ways: A 3 on the first roll and non 3 on other two rolls. This can be done in 1*5*5 = 25 ways. The 3 could be on the second or third roll too. So total favorable cases = 25*3 = 75 Required Probability = 75/216 = 25/72 Answer (C)
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Veritas Prep GMAT Instructor
Joined: 01 Jul 2017
Posts: 71
Location: United States
Concentration: Leadership, Organizational Behavior

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
06 Jan 2018, 17:32
Here is the full answer, from the ground up. To understand the mathematics behind this question we need to understand three basic probability rules: (1) A probability is calculated by dividing the number of options when the event occurs by the total number of possibilities.(2) When calculating the probability of multiple events that must ALL occur to meet a certain condition, we must multiply the separate probabilities of each event together. Thus, the probability of all the events \(A\), \(B\), and \(C\) occurring would be: \(P_{combined} = (P_A)*(P_B)*(P_C)\)(3) When calculating the probability of a series of mutuallyexclusive events, where ANY event occurring would be sufficient, we must add the separate probabilities of each event together. Thus, the probability of any of the events \(A\), \(B\), and \(C\) occurring would be: \(P_{combined} = (P_A)+(P_B)+(P_C)\)Let's leverage these three basic rules to solve this problem. First, the probability of rolling a single value (in this case 3) on a fair 6sided die would be one out of six (or \(\frac{1}{6}\)). The probability of NOT rolling that number would be five out of six (or \(\frac{5}{6}\)). So, if you need to roll a single value on one die, while simultaneously not rolling that value on the other two dice, then we invoke rule #2 from above (i.e., we multiply the probabilities.) Thus, \(P=(\frac{1}{6})*(\frac{5}{6})*(\frac{5}{6})=\frac{25}{216}\) Notice that this is answer choice A. However, this is a deliberate trap answer. With probabilities, the order always matters. (I like to call this "100% Chance of Order" in my classes.) Thus, if we don't care which of the dice roll the "3", then we need to look at each of the possible orders that could work. There are three possibilities: \(3, N, N\) \(N, 3, N\) \(N, N, 3\) The probability of each of these orders is the same (\(\frac{25}{216}\)), but each is separate and independent. We now need to invoke rule #3 from above and add the separate probabilities together: \(P_{Total} = \frac{25}{216} + \frac{25}{216} + \frac{25}{216} = \frac{3*25}{216}\) Recognizing common factors in the top and bottom of this fraction allow us to get to the answer without any messy math. \(216 = 6*6*6 = 2*3*6*6\). Thus, \(\frac{3*25}{216}=\frac{3*25}{2*3*6*6}=\frac{25}{2*6*6}=\frac{25}{72}\) The answer is C.
_________________
Aaron J. Pond Veritas Prep EliteLevel Instructor
Hit "+1 Kudos" if my post helped you understand the GMAT better. Look me up at https://www.veritasprep.com/gmat/aaronpond/ if you want to learn more GMAT Jujitsu.




Intern
Joined: 24 Sep 2015
Posts: 4

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
04 Nov 2015, 21:22
shyind wrote: If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled?
A. 25/216
B. 50/216
C. 25/72
D. 25/36
E. 5/6 Probability if getting 3 on a die is: P(3)=1/6 Probability of not getting 3 on a die is nP(3)=5/6 Let the three dies be denoted by P1,P2 and P3 Probability of getting 3 on the first die and not getting 3 on the other two dies is given by: =P1(3)*nP(3)*nP(3) =1/6*5/6*5/6 =25/216 As there are 3 dies,so similarly for the rest of the two dies probability will be 25/216 Summing up all the 3 cases we get: =3*25/216 =25/72



Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7086
GPA: 3.82

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
05 Nov 2015, 01:22
Forget conventional ways of solving math questions. In PS, IVY approach is the easiest and quickest way to find the answer. If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled? A. 25/216 B. 50/216 C. 25/72 D. 25/36 E. 5/6 Let O be represented as 3 being rolled, and X be represented as no 3 being rolled. Then the case of exactly one 3 being rolled can be represented as (O, X, X), (X, O, X), (X, X, O). The probability of (O, X, X) is (1/6)*(5/6)*(5/6), that of (X, O, X) is (5/6)*(1/6)*(5/6) and that of (X, X, O) is (5/6)*(5/6)*(1/6). So the probability is (1/6)*(5/6)*(5/6) + (5/6)*(1/6)*(5/6) + (5/6)*(5/6)*(1/6) = 3*(1/6)*(5/6)*(5/6) = 25/72. The answer is (C).
_________________
MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The oneandonly World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only $149 for 3 month Online Course" "Free Resources30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons  try it yourself"



Retired Moderator
Joined: 19 Mar 2014
Posts: 933
Location: India
Concentration: Finance, Entrepreneurship
GPA: 3.5

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
08 Jul 2017, 12:50
shyind wrote: If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled?
A. 25/216
B. 50/216
C. 25/72
D. 25/36
E. 5/6 Probability of getting 3 \(= \frac{1}{6}\) Probability of not getting 3 \(= \frac{5}{6}\) Probability of getting three on the first roll \(= \frac{1}{6} * \frac{5}{6} * \frac{5}{6}\) Probability of getting three on the first roll \(= \frac{25}{216}\) As we can get 3 on either second or the third time we will multiply the above probability by 3 \(= \frac{25}{216} * 3\) \(= \frac{25}{72}\) Hence, Answer is C
_________________
"Nothing in this world can take the place of persistence. Talent will not: nothing is more common than unsuccessful men with talent. Genius will not; unrewarded genius is almost a proverb. Education will not: the world is full of educated derelicts. Persistence and determination alone are omnipotent."
Best AWA Template: https://gmatclub.com/forum/howtoget60awamyguide64327.html#p470475



EMPOWERgmat Instructor
Status: GMAT Assassin/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 13755
Location: United States (CA)

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
07 Feb 2018, 12:46
Hi All, You can approach the math in this question in a couple of different ways, so you have to think about what would be the easiest way for YOU to organize your work. The questions tells us to roll a 6sided dice 3 times. We're asked for the probability of rolling EXACTLY one '3' on those three rolls. Here's a way to break the calculation down into 3 smaller calculations: (first roll is 3)(second roll is NOT 3)(third roll is NOT 3) = (1/6)(5/6)(5/6) = 25/216 (first roll is NOT 3)(second roll is 3)(third roll is NOT 3) = (5/6)(1/6)(5/6) = 25/216 (first roll is NOT 3)(second roll is NOT 3)(third roll is 3) = (5/6)(5/6)(1/6) = 25/216 Total = 3(25/216) = 75/216 = 25/72 Final Answer: GMAT assassins aren't born, they're made, Rich
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com
Rich Cohen
CoFounder & GMAT Assassin
Special Offer: Save $75 + GMAT Club Tests Free
Official GMAT Exam Packs + 70 Pt. Improvement Guarantee www.empowergmat.com/
*****Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!*****



Intern
Joined: 25 Nov 2017
Posts: 6

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
13 Apr 2018, 01:38
niel1989 wrote: shyind wrote: If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled?
A. 25/216
B. 50/216
C. 25/72
D. 25/36
E. 5/6 Probability if getting 3 on a die is: P(3)=1/6 Probability of not getting 3 on a die is nP(3)=5/6 Let the three dies be denoted by P1,P2 and P3 Probability of getting 3 on the first die and not getting 3 on the other two dies is given by: =P1(3)*nP(3)*nP(3) =1/6*5/6*5/6 =25/216 As there are 3 dies,so similarly for the rest of the two dies probability will be 25/216 Summing up all the 3 cases we get: =3*25/216 =25/72 "A" die is rolled 3 times.. There aren't 3 dies... which is why I can't seem to understand why not 25/216



Board of Directors
Status: Stepping into my 10 years long dream
Joined: 18 Jul 2015
Posts: 3619

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
13 Apr 2018, 01:44
saumya2805 wrote: "A" die is rolled 3 times.. There aren't 3 dies... which is why I can't seem to understand why not 25/216 Hey saumya2805 , There isn't any difference in rolling the same die thrice or rolling three different does. Hence, for each die we have the total outcomes = 6. Therefore, total possibilities= 6*6*6 = 216. Does that make sense?
_________________
My GMAT Story: From V21 to V40 My MBA Journey: My 10 years long MBA Dream My Secret Hacks: Best way to use GMATClub  Importance of an Error Log! Verbal Resources: All SC Resources at one place  All CR Resources at one place Blog: Subscribe to Question of the Day Blog GMAT Club Inbuilt Error Log Functionality  View More. New Visa Forum  Ask all your Visa Related Questions  here. New! Best Reply Functionality on GMAT Club! Find a bug in the new email templates and get rewarded with 2 weeks of GMATClub Tests for free Check our new About Us Page here.



Intern
Joined: 25 Nov 2017
Posts: 6

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
13 Apr 2018, 01:52
EMPOWERgmatRichC wrote: Hi All, You can approach the math in this question in a couple of different ways, so you have to think about what would be the easiest way for YOU to organize your work. The questions tells us to roll a 6sided dice 3 times. We're asked for the probability of rolling EXACTLY one '3' on those three rolls. Here's a way to break the calculation down into 3 smaller calculations: (first roll is 3)(second roll is NOT 3)(third roll is NOT 3) = (1/6)(5/6)(5/6) = 25/216 (first roll is NOT 3)(second roll is 3)(third roll is NOT 3) = (5/6)(1/6)(5/6) = 25/216 (first roll is NOT 3)(second roll is NOT 3)(third roll is 3) = (5/6)(5/6)(1/6) = 25/216 Total = 3(25/216) = 75/216 = 25/72 Final Answer: GMAT assassins aren't born, they're made, Rich Hi, Can you please help me? I completely understand the logic you've followed... What I don't understand is why the need.. I mean, the question simply asks the probability of 3 appearing exactly once. It doesn't specify to find for "on which of these throws..." There's a probability of 3 appearing exactly once, 25 out of 216 times. If the question had been that 3 dies are rolled simultaneously 3 times... it would've made more sense to me.. Since you're an expert, surely I'm missing something. Could you please help me understand what I'm missing?



Intern
Joined: 25 Nov 2017
Posts: 6

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
13 Apr 2018, 02:55
AaronPond wrote: Here is the full answer, from the ground up. To understand the mathematics behind this question we need to understand three basic probability rules:
(1) A probability is calculated by dividing the number of options when the event occurs by the total number of possibilities.
(2) When calculating the probability of multiple events that must ALL occur to meet a certain condition, we must multiply the separate probabilities of each event together. Thus, the probability of all the events \(A\), \(B\), and \(C\) occurring would be: \(P_{combined} = (P_A)*(P_B)*(P_C)\)
(3) When calculating the probability of a series of mutuallyexclusive events, where ANY event occurring would be sufficient, we must add the separate probabilities of each event together. Thus, the probability of any of the events \(A\), \(B\), and \(C\) occurring would be: \(P_{combined} = (P_A)+(P_B)+(P_C)\)
Let's leverage these three basic rules to solve this problem.
First, the probability of rolling a single value (in this case 3) on a fair 6sided die would be one out of six (or \(\frac{1}{6}\)).
The probability of NOT rolling that number would be five out of six (or \(\frac{5}{6}\)).
So, if you need to roll a single value on one die, while simultaneously not rolling that value on the other two dice, then we invoke rule #2 from above (i.e., we multiply the probabilities.) Thus,
\(P=(\frac{1}{6})*(\frac{5}{6})*(\frac{5}{6})=\frac{25}{216}\)
Notice that this is answer choice A. However, this is a deliberate trap answer. With probabilities, the order always matters. (I like to call this "100% Chance of Order" in my classes.) Thus, if we don't care which of the dice roll the "3", then we need to look at each of the possible orders that could work. There are three possibilities:
\(3, N, N\) \(N, 3, N\) \(N, N, 3\)
The probability of each of these orders is the same (\(\frac{25}{216}\)), but each is separate and independent. We now need to invoke rule #3 from above and add the separate probabilities together:
\(P_{Total} = \frac{25}{216} + \frac{25}{216} + \frac{25}{216} = \frac{3*25}{216}\)
Recognizing common factors in the top and bottom of this fraction allow us to get to the answer without any messy math. \(216 = 6*6*6 = 2*3*6*6\). Thus,
\(\frac{3*25}{216}=\frac{3*25}{2*3*6*6}=\frac{25}{2*6*6}=\frac{25}{72}\)
The answer is C. You mention the below: "Notice that this is answer choice A. However, this is a deliberate trap answer. With probabilities, the order always matters. (I like to call this "100% Chance of Order" in my classes.) Thus, if we don't care which of the dice roll the "3", then we need to look at each of the possible orders that could work." Plz refer the bold colored portion.. Is it understood by default, that the order always matters? Even if it isn't mentioned explicitly in the question? This is exactly what I'm always confused about in probability related questions.



Veritas Prep GMAT Instructor
Joined: 01 Jul 2017
Posts: 71
Location: United States
Concentration: Leadership, Organizational Behavior

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
13 Apr 2018, 15:43
saumya2805 wrote: You mention the below: "Notice that this is answer choice A. However, this is a deliberate trap answer. With probabilities, the order always matters. (I like to call this "100% Chance of Order" in my classes.) Thus, if we don't care which of the dice roll the "3", then we need to look at each of the possible orders that could work."
Plz refer the bold colored portion.. Is it understood by default, that the order always matters? Even if it isn't mentioned explicitly in the question? This is exactly what I'm always confused about in probability related questions. Thank you, Saumya2805, for the question. Yes, when calculating the probability of multiple situations, the order always matters. This isn't just the "default", this is how the mathematics of probability work. Even if the events are simultaneous, think about the solution one event at a time. Each different arrangement has a separate probability.
_________________
Aaron J. Pond Veritas Prep EliteLevel Instructor
Hit "+1 Kudos" if my post helped you understand the GMAT better. Look me up at https://www.veritasprep.com/gmat/aaronpond/ if you want to learn more GMAT Jujitsu.



Intern
Joined: 25 Nov 2017
Posts: 6

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
14 Apr 2018, 00:24
AaronPond wrote: saumya2805 wrote: You mention the below: "Notice that this is answer choice A. However, this is a deliberate trap answer. With probabilities, the order always matters. (I like to call this "100% Chance of Order" in my classes.) Thus, if we don't care which of the dice roll the "3", then we need to look at each of the possible orders that could work."
Plz refer the bold colored portion.. Is it understood by default, that the order always matters? Even if it isn't mentioned explicitly in the question? This is exactly what I'm always confused about in probability related questions. Thank you, Saumya2805, for the question. Yes, when calculating the probability of multiple situations, the order always matters. This isn't just the "default", this is how the mathematics of probability work. Even if the events are simultaneous, think about the solution one event at a time. Each different arrangement has a separate probability. Thanks Aaron, that helps!



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8982
Location: Pune, India

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
15 Apr 2018, 18:15
saumya2805 wrote: niel1989 wrote: shyind wrote: If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled?
A. 25/216
B. 50/216
C. 25/72
D. 25/36
E. 5/6 Probability if getting 3 on a die is: P(3)=1/6 Probability of not getting 3 on a die is nP(3)=5/6 Let the three dies be denoted by P1,P2 and P3 Probability of getting 3 on the first die and not getting 3 on the other two dies is given by: =P1(3)*nP(3)*nP(3) =1/6*5/6*5/6 =25/216 As there are 3 dies,so similarly for the rest of the two dies probability will be 25/216 Summing up all the 3 cases we get: =3*25/216 =25/72 "A" die is rolled 3 times.. There aren't 3 dies... which is why I can't seem to understand why not 25/216 How do you get 216? You say the FIRST roll could be 1/2/3/4/5/6 i.e. 6 ways The SECOND roll could be 1/2/3/4/5/6 i.e. 6 ways The THIRD roll could be 1/2/3/4/5/6 i.e. 6 ways Hence rolling a 3 on FIRST roll and any of the other 5 numbers of other two rolls is different from rolling a 3 on SECOND roll and any of the other 5 numbers on other two rolls because we have counted these cases apart {3, 1, 6}, {1, 3, 6}. That is how we get 75 and not just 25. This is same as saying  I have 3 dice of different colours  Red, Yellow and Blue  and I roll them together The Red could have outcome in 6 ways, Yellow in 6 and Blue in 6 so 216 ways Same as case above. How about saying I have 3 identical dice and I roll them together. How many outcomes do I have in that case? Is (1, 2, 3) different from (3, 1, 2)? No. Then I do not have 216 outcomes in this case. In this case, there is no "order".
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Senior Manager
Joined: 22 Sep 2018
Posts: 250

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
25 Jan 2019, 16:31
how would you solve this question using combinations?



Director
Joined: 04 Sep 2015
Posts: 556
Location: India
WE: Information Technology (Computer Software)

Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
Show Tags
22 Feb 2019, 12:05
If a fair 6sided die is rolled three times, what is the probability that exactly one 3 is rolled?
A. 25/216
B. 50/216
C. 25/72
D. 25/36
E. 5/6
Solution: prob of getting 3=1/6;prob of not getting 3=5/6
exactly one 3=(5/6*5/6*1/6)*3 =25/72




Re: If a fair 6sided die is rolled three times, what is the probability
[#permalink]
22 Feb 2019, 12:05






