# A game of probability

An unbiased die having the numbers 1,2,3,4,5,6 is rolled 4 times. What is the probability that the minimum face value is 2?

According to my reasoning the answer should be $$\frac{5^4}{6^4}$$ because we have the options 2,3,4,5 and 6 . Which can be chosen for four trials in $$5^4$$ ways .

However the correct answer seems to be $$\frac{5^4-4^4}{6^4}$$ and I can’t reason it out. Please help me where I’m missing out

Thank you !

• $(5/6)^4$ is the probability that the minimum face value is $\ge 2$. – ncmathsadist Oct 19 '18 at 17:34

There are $$5^4$$ roll sequences formed from the numbers 2, 3, 4, 5, 6 (and thus having minimum digit at least 2). Of these, $$4^4$$ roll sequences are formed from 3, 4, 5, 6, so cannot have minimum digit 2. The remaining $$5^4-4^4$$ sequences thus have minimum digit exactly 2, as required, and dividing by the $$6^4$$ rolls in total yields the correct probability.

• Ohhh I didn’t know that it was strict to have 2 in the answer . They never mentioned that $2$ must be one of the outcomes though so I was a bit confused. How do you deduce that $2$ must be one of the outcomes ? – Aditi Oct 19 '18 at 17:38
• @Aditi The minimum face value is stipulated to be 2. This means that at least one die actually shows 2, to realise the minimum. – Parcly Taxel Oct 19 '18 at 17:38
• Ohh now I get it ! Thank you :) – Aditi Oct 19 '18 at 17:39

Hint:

Your answer would be good if the question were "What is the probability that the minimum face is not $$1$$"?

• Thank you , but how do we deduce that 2 must be one of the outcomes ? – Aditi Oct 19 '18 at 17:38
• You can't. Every roll could be fives and sixes (for example). There it is the point. – ajotatxe Oct 19 '18 at 17:39
• Thank you for the hint ! – Aditi Oct 19 '18 at 17:43

You can also calculate the numerator directly: $${4 \choose 1}*4^3$$ sequences have a single 2; $${4 \choose 2}*4^2$$ have two 2's; $${4 \choose 3}*4^1$$sequences have 3 2's; 1 sequence of 2222 (all sequences containing no 1's). If you add these up, you get $$369=5^4-4^4$$.