# A die is rolled 4 times .What is the probability of getting only one 6?

1
by chityin

2014-09-21T20:52:44+05:30

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Probability of getting a six in one throw / roll :  1/6

We get one 6 only in four rolls or throws of the die.  This means that the other three throws we get other than 6.
Probability of getting a non-six (from 1 to 5) in one throw or roll : 5/6

Throwing of a die first time is independent from other throws. So probabilities multiply.

You can get one 6 in the first throw, or 2nd throw, or 3rd throw or the fourth roll. So,

Probability = 1/6 * 5/6 * 5/6 * 5/6   +   5/6 * 1/6 * 5/6 * 5/6
+   5/6 * 5/6 * 1/6 * 5/6  +   5/6 * 5/6 * 5/6 * 1/6
= 4 * 1* 5³ / 6⁴ = 0.3858

= 38.58 %
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For those who know binomial expansion :

(p + q)⁴ = gives probabilities for occurrences of 6 or other than 6 when a die is rolled 4 times.  p = 1/6   q = 5/6.   The second term in the expansion gives:

⁴C₁ p q³ =  ⁴C₁ (1/6) (5/6)³ = 0.3858