A fair six-sided die with faces numbered 1 through 6 is rolled twice. What is the probabil… — GRE General Test

A fair six-sided die with faces numbered $1$ through $6$ is rolled twice. What is the probability of rolling a $4$ on the first roll and an odd number on the second roll?

$\frac{1}{12}$ Cevap
B
$\frac{2}{3}$
C
$\frac{1}{3}$
D
$\frac{1}{4}$
E
$\frac{1}{36}$

Cevap

\frac{1}{12}

Because the outcome of the first die roll does not affect the outcome of the second die roll, the two events are independent. The probability of rolling a

4

on the first roll is

\frac{1}{6}

, and the probability of rolling an odd number (

1, 3,

5

) on the second roll is

\frac{3}{6} = \frac{1}{2}

. Multiplying these individual probabilities together yields

\frac{1}{6} \times \frac{1}{2} = \frac{1}{12}

Adım Adım Çözüm

Determine the probability of the first event (rolling a 4).

There is

1

favorable outcome out of

6

possible outcomes, so

P(\text{First roll is } 4) = \frac{1}{6}

Each face of a fair six-sided die is equally likely to land face up.

Determine the probability of the second event (rolling an odd number).

The odd numbers are

1, 3,

and

5

, giving

3

favorable outcomes out of

6

total outcomes, so

P(\text{Second roll is odd}) = \frac{3}{6} = \frac{1}{2}

Half of the outcomes on a standard six-sided die are odd.

Calculate the joint probability of both independent events occurring.

P(\text{4 on first AND odd on second}) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12}

For independent events

A

and

B

, the multiplication rule states that

P(A \text{ and } B) = P(A) \times P(B)

Anahtar Kavram

Probability of Independent Events

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Soru

Cevap

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