For one roll of a certain die, the probability of rolling a two is \(1/6\). If this die is rolled \(4\) times, which of

[BTGmoderatorLU]

Source: Manhattan Prep

For one roll of a certain die, the probability of rolling a two is \(1/6\). If this die is rolled \(4\) times, which of the following is the probability that the outcome will be a two at least \(3\) times?

A. \(\left(\dfrac{1}{6}\right)^4\)

B. \(2\left(\dfrac{1}{6}\right)^3+\left(\dfrac{1}{6}\right)^4\)

C. \(3\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)+\left(\dfrac{1}{6}\right)^4\)

D. \(4\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)+\left(\dfrac{1}{6}\right)^4\)

E. \(6\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)+\left(\dfrac{1}{6}\right)^4\)

The OA is D

[Asimo3089]

Source: Manhattan Prep

For one roll of a certain die, the probability of rolling a two is \(1/6\). If this die is rolled \(4\) times, which of the following is the probability that the outcome will be a two at least \(3\) times?

A. \(\left(\dfrac{1}{6}\right)^4\)

B. \(2\left(\dfrac{1}{6}\right)^3+\left(\dfrac{1}{6}\right)^4\)

C. \(3\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)+\left(\dfrac{1}{6}\right)^4\)

D. \(4\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)+\left(\dfrac{1}{6}\right)^4\)

E. \(6\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)+\left(\dfrac{1}{6}\right)^4\)

The OA is D

Dice roll, two chance \(1/6\), four times. Find chance of getting two at least three times.

To solve, we choose option with two three times, any outcome on fourth roll, and also possibility of two four times. Correct option: D.

Mathematically, we have \(4\left(\dfrac{1}{6}\right)^3\left(\dfrac{5}{6}\right)+\left(\dfrac{1}{6}\right)^4\) as the chance of getting two at least three times out of four rolls.