Problem 41 - A First Course in Probability

Question

If a die is rolled 4 times, what is the probability that 6 comes up at least once?

Step-by-Step Solution

Answer

The probability of getting at least one 6 in four rolls of a die is \(\frac{671}{1296}\).

Step 1: Determine the probability of not getting a 6 in one roll

Since a fair die has 6 faces, the possibility of not getting a 6 in one roll is the probability of getting any of the other 5 faces. Therefore, the probability of not getting a 6 in one roll is 5/6.

Step 2: Determine the probability of not getting a 6 in all four rolls

The probability of not getting a 6 in all four rolls can be found by multiplying the probability of not getting a 6 in one roll by itself four times. Mathematically, this can be expressed as \((\frac{5}{6})^4\).

Step 3: Calculate the probability of not getting a 6 in all four rolls

Using the formula from step 2, we calculate the probability of not getting a 6 in all four rolls: \((\frac{5}{6})^4 = \frac{625}{1296}\).

Step 4: Calculate the probability of getting at least one 6 in the four rolls

Now, to find the probability of getting at least one 6 in the four rolls, we need to subtract the probability of not getting a 6 in all four rolls from 1: \(1 - \frac{625}{1296} = \frac{671}{1296}\).

Step 5: Simplify the probability

The probability of getting at least one 6 in the four rolls is \(\frac{671}{1296}\). This fraction is already in its simplest form, so the final answer is \(\boxed{\frac{671}{1296}}\).