The probabilities that drivers A, B and C will each drive home safely after consuming liquor are 2/5, 3/7 and 3/4 respectively. Assuming their driving events are independent, what is the probability that all three drivers drive home safely?

Introduction / Context:
This question illustrates how to combine independent probabilities for several people performing the same type of action. Each driver has a given probability of reaching home safely after consuming liquor, and the problem asks for the chance that all three do so. This is a straightforward application of the multiplication rule for independent events.

Given Data / Assumptions:

• Probability that driver A drives home safely = 2/5.
• Probability that driver B drives home safely = 3/7.
• Probability that driver C drives home safely = 3/4.
• The driving results of A, B and C are independent of each other.
• Event of interest: all three drivers reach home safely.

Concept / Approach:
When events are independent, the probability that they all occur together is the product of their individual probabilities. Here, we multiply the probabilities for A, B and C each driving safely. There is no need to consider any order, because the events occur simultaneously and independently, and the multiplication rule captures this.

Step-by-Step Solution:
Probability that A is safe = 2/5. Probability that B is safe = 3/7. Probability that C is safe = 3/4. Required probability that all three are safe = (2/5) * (3/7) * (3/4). First multiply numerators: 2 * 3 * 3 = 18. Multiply denominators: 5 * 7 * 4 = 140. So probability = 18 / 140. Simplify 18 / 140 by dividing numerator and denominator by 2 to get 9 / 70.

Verification / Alternative check:
We can confirm the result by simplifying at an earlier step. For example, write (2/5) * (3/7) * (3/4) and cancel a factor of 2 from 2 and 4 to get (1/5) * (3/7) * (3/2). Then multiply 1 * 3 * 3 = 9 and 5 * 7 * 2 = 70, giving 9 / 70 again. This double check shows that our simplification is consistent and that no arithmetic error has been made.

Why Other Options Are Wrong:
3/70 and 4/70 are smaller fractions that would imply a much lower probability than is justified by the given numbers. 1/50 does not correspond to any natural simplification of the product of 2/5, 3/7 and 3/4.

Common Pitfalls:
A frequent misunderstanding is to add probabilities instead of multiplying them when dealing with independent events. Another mistake is to use incorrect denominators by multiplying only some of the terms or by failing to simplify properly. Carefully multiplying all three fractions and simplifying step by step avoids these issues and leads to the correct result.

Final Answer:
The probability that all three drivers will drive home safely is 9/70.