You roll a fair die three times. what is the probability of each of the following?
a) you roll all 4's.
b) you roll all even numbers.
c) none of your rolls gets a number divisible by 2.
d) you roll at least one 2.
e) the numbers you roll are not all 2's.

(a)

(b)

(c)

(d)

(e)

Step-by-step explanation:

It is given that we roll a fair die three times. We need to find the following probability.

Total possible value = 1,2,3,4,5,6

(a) You roll all 4's.

Probability of getting a 4 =

Probability of getting 4 on all rolls =

Therefore, the probability of getting 4 on all rolls is .

(b) You roll all even numbers.

Even number = 2,4,6

Probability of getting an even number

Probability of getting even number on all rolls =

Therefore, the probability of getting even number on all rolls is .

(c) None of your rolls gets a number divisible by 2.

odd number = 1,3,5

Probability of getting an odd number

Probability of getting odd number on all rolls =

Therefore, the probability of getting odd number on all rolls is .

(d) You roll at least one 2.

Probability of getting a 2 =

Probability of getting any number except 2 =

Probability of getting any number except 2 in all rolls =

Probability of getting at least one 2 = 1 - P(getting no 2)

Probability of getting at least one 2 =

Therefore, the probability of getting at least one 2 is .

(e) The numbers you roll are not all 2's.

Probability of getting a 2 =

Probability of getting all 2 =

Probability of getting all not all 2's =

Therefore, the probability of getting not all 2's is .

answer: all of the odd numbers were opened

the first person was the one that was opening all of the lockers. since one is odd, this causes all of the odd number lockers to be opened in the end.