A quality inspection bin contains 6 components manufactured by Line 1 and 4 components man… — GRE General Test

A quality inspection bin contains $6$ components manufactured by Line 1 and $4$ components manufactured by Line 2. Two components are drawn randomly from the bin sequentially, without replacement. Let $B$ be the event that the second component drawn is manufactured by Line 1, and let $C$ be the event that at least one of the two components drawn is manufactured by Line 2. What is the conditional probability $P(B \mid C)$ ?

$\frac{2}{5}$ Cevap
B
$\frac{4}{15}$
C
$\frac{3}{5}$
D
$\frac{4}{9}$
E
$\frac{2}{3}$

Cevap

The conditional probability

P(B \mid C)

\frac{2}{5}

To find

P(B \mid C)

, we evaluate the ratio

\frac{P(B \cap C)}{P(C)}

. The probability of event

C

(at least one component from Line 2) is most easily found by taking the complement of drawing two Line 1 components:

1 - \frac{6 \times 5}{10 \times 9} = \frac{60}{90}

. For event

B \cap C

to occur, the second component must be Line 1 and at least one component must be Line 2, meaning the sequence must be (Line 2, Line 1), which has probability

\frac{4 \times 6}{10 \times 9} = \frac{24}{90}

. Dividing

\frac{24}{90}

\frac{60}{90}

yields

\frac{24}{60} = \frac{2}{5}

Adım Adım Çözüm

Calculate the total number of outcomes for drawing two components sequentially without replacement.

Total outcomes =

10 \times 9 = 90

There are 10 components available for the first selection and 9 remaining components for the second selection.

Determine the probability of event

C

(at least one component from Line 2) using the complement rule.

P(C) = 1 - P(\text{both from Line 1}) = 1 - \frac{6 \times 5}{90} = 1 - \frac{30}{90} = \frac{60}{90} = \frac{2}{3}

The complement of having at least one component from Line 2 is having both components drawn from Line 1.

Determine the probability of the joint event

B \cap C

P(B \cap C) = P(\text{first from Line 2 AND second from Line 1}) = \frac{4 \times 6}{90} = \frac{24}{90} = \frac{4}{15}

For event

B

(second is Line 1) and event

C

(at least one is Line 2) to occur simultaneously, the first component must be from Line 2 and the second from Line 1.

Apply the conditional probability formula

P(B \mid C) = \frac{P(B \cap C)}{P(C)}

P(B \mid C) = \frac{24/90}{60/90} = \frac{24}{60} = \frac{2}{5}

The conditional probability isolates the probability of event

B

within the reduced sample space where event

C

has occurred.

Anahtar Kavram

Conditional Probability and Dependent Sequential Events

Tahmini Süre:2m 30s

Soru

Cevap

Adım Adım Çözüm

Anahtar Kavram

Sık Yapılan Hatalar

Konular

Bu konudan daha fazla soru