What is the probability of getting the number 6 at least twice in a regular dice if I can roll it 4 times?

This post is about an aptitude question which is asked in the GATE (Graduate Aptitude Test in Engineering).

The question is that “What is probability of occurring SIX for at least two times if you roll dice for four times?”

·        Now He is asking you to find probability for at least two times. So you need to find probability of occurring SIX for two times, three times and four times. But it will be more easy if you just t find probability of occurring SIX for zero times and for one time and deduct that amount from 1 and you will get desired answer.

·        Now If you roll a dice once that probability of occurring SIX is 1/6 and probability of not occurring is 5/6 . Here you are rolling dice for 4 times so Probability of not occurring SIX will be [(5/6)* (5/6)* (5/6)* (5/6)]. This shows us probability of occurring SIX zero time as per we needed. Now let’s move further

·        Now for probability of occurring SIX for once. When you roll the dice for four times then there should be one roll such way that you can get SIX in that roll. Selection of this roll among all four rolls can be give as    ⁴C₁. Now Probability of occurring SIX in that selected roll is 1/6 but here important thing is that there are also other three rolls in which SIX is not occurring. So Probability of occurring SIX for once can be written as :

⁴C₁ (1/6) [(5/6)*(5/6)*(5/6)]

So basically,

Probability of occurring SIX at least for two times

In the four rolls of a fine dice 

= 1- Probability of occurring Six     for

                                                            Once or probability of occurring Six    

                                                            For zero times

= 1 – {[(5/6)* (5/6)*(5/6)* (5/6)] + ⁴C₁ (1/6) [(5/6)*(5/6)*(5/6)]}

= 171/1296

=19/144