A number is chosen at random from the numbers 10 to 99. By seeing the number a man will laugh if product of the digits is 12. If he choose three numbers with replacement then the probability that he will laugh at least once is:
Step-by-Step Solution
Key Concept: Identify two-digit numbers whose digits multiply to 12, then use the complement rule for the probability of at least one success in three trials.
The numbers whose digit product is 12 are: 43, 34, 62, 26 (four numbers). The probability man laughs is $\frac{4}{90} = \frac{2}{45}$. The required probability that he does NOT laugh is $1 - \left(1 - \frac{2}{45}\right)^3 = 1 - \left(\frac{43}{45}\right)^3$.
Correct Answer: 4