Mathematics: Aweighted die, numbered one through six, has a probability of 1/4 of rolling a six. if this die is rolled three times, and each roll is independent, what is the probability of rolling at least two sixes?

what are the answersStep-by-step explanation:...

Aweighted die, numbered one through six, has

Ответ:

journeyhile5

15.03.2022

Mathematics

$\frac{1}{4}$

Step-by-step explanation:

Three events A, B and C are said to be independent if occurrence of one outcome does not affect the other .

In such case P ( A and B and C ) = P(A) × P(B) × P (C)

Given : A weighted die, numbered one through six, has a probability of 1/4 of rolling a six .

As the die is rolled three times, possible outcomes are $\left \{ \left ( 6,6,1 \right )\,,\,\left ( 6,6,2 \right )\,,\,\left ( 6,6,3 \right )\,,\,\left ( 6,6,4 \right )\,,\,\left ( 6,6,5 \right )\,,\,\left ( 6,6,6 \right ) \right \}$

Let E denotes the event: 6 occurs

Let F denotes the event that 6 does not occur .

Given: P(E) = $\frac{1}{4}$

So, P(F) = $1-\frac{1}{4}=\frac{3}{4}$

Therefore, probability of rolling at least two sixes = $P\left ( 6,6,1 \right )+P\left ( 6,6,2 \right )+P\left ( 6,6,3 \right )+P\left ( 6,6,4 \right )+P\left ( 6,6,5 \right )+P\left ( 6,6,6 \right )$

$=\left ( \frac{1}{4} \right )\left ( \frac{1}{4} \right )\left ( \frac{3}{4} \right )+\left ( \frac{1}{4} \right )\left ( \frac{1}{4} \right )\left ( \frac{3}{4} \right )+\left ( \frac{1}{4} \right )\left ( \frac{1}{4} \right )\left ( \frac{3}{4} \right )+\left ( \frac{1}{4} \right )\left ( \frac{1}{4} \right )\left ( \frac{3}{4} \right )+\left ( \frac{1}{4} \right )\left ( \frac{1}{4} \right )\left ( \frac{3}{4} \right )+\left ( \frac{1}{4} \right )\left ( \frac{1}{4} \right )\left ( \frac{1}{4} \right )$

$=\frac{3}{64}+\frac{3}{64}+\frac{3}{64}+\frac{3}{64}+\frac{3}{64}+\frac{1}{64}\\\\=\frac{16}{64}\\\\=\frac{1}{4}$

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Aweighted die, numbered one through six, has a probability of 1/4 of rolling a six. if this die is rolled three times, and each roll is independent, what is the probability of rolling at least two sixes?

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