A financial firm has a team of 10 auditors consisting of 6 certified public accountants (C… — GMAT

A financial firm has a team of 10 auditors consisting of 6 certified public accountants (CPAs) and 4 audit assistants. A special audit task force of 3 auditors is to be selected at random from the team. If the probability that the task force contains at least one CPA is expressed as a fraction $\frac{a}{b}$ in simplest form, what is the value of $a + b$ ?

Answer: 59

Answer

To find the probability of selecting at least one CPA, it is most efficient to use the complementary probability formula:

P(\text{at least 1 CPA}) = 1 - P(\text{no CPAs})

. The total number of ways to select any 3 auditors from the team of 10 is given by

\binom{10}{3} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120

. The number of ways to select 3 auditors such that none are CPAs (i.e., all 3 are audit assistants) is

\binom{4}{3} = 4

. Therefore, the probability of choosing zero CPAs is

\frac{4}{120} = \frac{1}{30}

. Subtracting from 1 gives

1 - \frac{1}{30} = \frac{29}{30}

. The fraction

\frac{29}{30}

is in simplest form because 29 is a prime number and does not divide 30. Thus,

a = 29

and

b = 30

, making

a + b = 59

Step-by-Step Solution

Calculate total possible combinations for selecting 3 members from 10.

\binom{10}{3} = 120

Selection is made without replacement and order of selection does not matter.

Calculate combinations of selecting 3 members with no CPAs.

\binom{4}{3} = 4

All 3 selected members must come from the pool of 4 audit assistants.

Determine probability of the complement event (no CPAs).

P(\text{no CPA}) = \frac{4}{120} = \frac{1}{30}

Probability is favorable outcomes over total outcomes.

Compute probability of at least one CPA.

P(\text{at least 1 CPA}) = 1 - \frac{1}{30} = \frac{29}{30}

P(\text{at least one}) = 1 - P(\text{none}).

Find the sum of numerator a and denominator b in simplest form.

a + b = 29 + 30 = 59

29/30 cannot be simplified further as 29 is a prime number.

Key Concept

Complementary Probability and At-Least-One Scenarios

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