An executive is monitoring two independent corporate projects, Project Alpha and Project B… — GRE General Test

An executive is monitoring two independent corporate projects, Project Alpha and Project Beta. Based on historical performance, the probability that Project Alpha meets its deadline is $\frac{4}{5}$ , and the probability that Project Beta meets its deadline is $\frac{3}{4}$ . What is the probability that exactly one of the two projects meets its deadline?

$\frac{7}{20}$ Cevap
B
$\frac{3}{5}$
C
$\frac{19}{20}$
D
$\frac{7}{9}$
E
$\frac{1}{20}$

Cevap

The probability that exactly one project meets its deadline is

\frac{7}{20}

The event 'exactly one project meets its deadline' consists of two mutually exclusive scenarios: (1) Alpha meets its deadline and Beta does not, or (2) Alpha misses its deadline and Beta meets it. Using independence, the probability of Scenario 1 is

\frac{4}{5} \times \left(1 - \frac{3}{4}\right) = \frac{4}{5} \times \frac{1}{4} = \frac{4}{20}

. The probability of Scenario 2 is

\left(1 - \frac{4}{5}\right) \times \frac{3}{4} = \frac{1}{5} \times \frac{3}{4} = \frac{3}{20}

. Summing these mutually exclusive probabilities gives

\frac{4}{20} + \frac{3}{20} = \frac{7}{20}

Adım Adım Çözüm

Determine the probabilities of individual events and their complements.

P(\text{Alpha meets}) = \frac{4}{5}

P(\text{Alpha misses}) = 1 - \frac{4}{5} = \frac{1}{5}

P(\text{Beta meets}) = \frac{3}{4}

P(\text{Beta misses}) = 1 - \frac{3}{4} = \frac{1}{4}

To find the probability of specific outcome combinations, the complementary probabilities for each independent event are required.

Identify the mutually exclusive cases that satisfy the condition 'exactly one project meets its deadline'.

Case 1: Alpha meets and Beta misses. Case 2: Alpha misses and Beta meets.

The event 'exactly one' consists of two distinct, non-overlapping scenarios.

Calculate the joint probability for each case using independence.

Case 1 probability:

\frac{4}{5} \times \frac{1}{4} = \frac{4}{20}

. Case 2 probability:

\frac{1}{5} \times \frac{3}{4} = \frac{3}{20}

Since the projects operate independently, joint probabilities are found by multiplying individual event probabilities.

Add the probabilities of the mutually exclusive cases.

\frac{4}{20} + \frac{3}{20} = \frac{7}{20}

For mutually exclusive events, the total probability of either case occurring is the sum of their individual probabilities.

Anahtar Kavram

Probability of Independent Events and Mutually Exclusive Cases

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