The newest invention of the 6.431x staff is a three-sided die. On any roll of this die, the result is 1 with probability 1/2, 2 with probability 1/4, and 3 with probability 1/4. Consider a sequence of six independent rolls of this die. Find the probability that exactly two of the rolls result in a 3.

Step-by-step explanation:

The given die is a three sided die, hence, there are only three possibilities of getting the outcomes.

We need to find the probability of getting exactly 3s as the result.

From the sequence of 6 independent rolls, 2 rolls can be chosen in ways.

The probability of getting two 3 as outcome is .

In the rest of the 4 sequences, will not be any 3 as outcome.

Probability of not getting a outcome rather than 3 is .

Hence, the required probability is ≅0.2966 or, 0.23.

the dimensions are:

step-by-step explanation:

we want to find the dimensions of a rectangular box with volume of

we factor to obtain;

we rewrite the rightmost factor to obtain;

recall that:

we apply difference of two squares formula to obtain;

hence the dimensions are