A fair 6-sided die is rolled 5 times. Find the probability of getting at least three 6s.

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Step-by-Step Solution

Step 1: Identify the type of probability distribution

This problem involves a fixed number of trials ($n=5$), where each trial has only two possible outcomes (getting a 6 or not getting a 6), and the probability of success ($p$) is constant for each trial. This indicates a binomial distribution.

Step 2: Determine parameters for the binomial distribution

The number of trials, $n$, is 5 (since the die is rolled 5 times). The probability of success, $p$, is the probability of rolling a 6 on a fair 6-sided die, which is $1/6$. Consequently, the probability of failure, $1-p$, is $5/6.

Step 3: Calculate probabilities for at least three 6s

We need to find the probability of getting at least three 6s, which means $P(X \ge 3)$. This is the sum of probabilities of getting exactly three 6s, exactly four 6s, and exactly five 6s. We use the binomial probability formula for each case.

Step 4: Compute individual probabilities

Now we compute the values for each term. $\binom{5}{3} = 10$, $\binom{5}{4} = 5$, and $\binom{5}{5} = 1$. We then multiply these combinations by the respective powers of $p$ and $(1-p)$. Note that $6^5 = 7776.

Step 5: Sum the probabilities

Finally, we sum the calculated probabilities for $P(X=3)$, $P(X=4)$, and $P(X=5)$ to get the total probability of getting at least three 6s.

Step 6: Simplify the result

The fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor. Dividing by 4 gives $69/1944$, and then dividing by 3 gives $23/648.