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
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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
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Dice roll, two chance \(1/6\), four times. Find chance of getting two at least three times.BTGmoderatorLU wrote: ↑Sat Jun 10, 2023 3:04 pmSource: 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
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.
