Three dice are thrown together. Find the probability of getting a total of at least 6.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\frac{103}{108}\)
“At least 6” covers every total except 3, 4 and 5, so it is quicker to count the complement (a total of 5 or less) and subtract its probability from 1. Three dice give \(6\times6\times6=216\) equally likely outcomes.
Ways of getting a total of 5 or less:
Total 3: (1, 1, 1), 1 way
Total 4: one die shows 2 and the others 1, 3 ways
Total 5: (1, 1, 3) in 3 orders and (1, 2, 2) in 3 orders, 6 ways
Totals of 5 or less altogether: \(1+3+6=10\) ways
Probability of a total of at least 6 \(\displaystyle =1-\frac{10}{216}=\frac{206}{216}=\frac{103}{108}\)
Option (1), \(\frac{5}{108}=\frac{10}{216}\), is the probability of the complement, a total of 5 or less.