Question

The probabilities of a boy passing English and Mathematics tests are a and b respectively. Find the probability of the boy failing both tests.

Answer 1

Answer on Question #55356 – Math – Statistics and Probability

Question

The probabilities of a boy passing English and Mathematics tests are a and b respectively. Find the probability of the boy failing both tests.

Solution

If probability of a boy passing English test is

$$Prob(English\ pass) = a,$$

then probability of a boy failing English test is

$$Prob(English\ fail) = 1 - Prob(English\ pass) = 1 - a,$$

because it is always the case that either boy passes the test or fails it.

Thus, by the complement rule for probability,

$$Prob(English\ pass) + Prob(English\ fail) = 1.$$

Similarly, if probability of a boy passing Mathematics test is

$$Prob(Mathematics\ pass) = b,$$

then probability of a boy failing Mathematics test is

$$Prob(Mathematics\ fail) = 1 - Prob(Mathematics\ pass) = 1 - b.$$

So, according to the multiplication rule of probability for independent events,

the probability of failing both test is

$$\begin{array}{l} Prob(both\ fail) = Prob(English\ fail\ and\ Mathematics\ fail) = \\ = Prob(English\ fail) \cdot Prob(Mathematics\ fail) = (1 - a)(1 - b). \end{array}$$

This is true, because events “English fail” and “Mathematics fail” are independent (no event implies or influences the result of the other).

Answer: $(1 - a)(1 - b)$.

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