Let A be the event that there are at least two fives among the four rolls. Let B be the event that there is at most one five among the four rolls. Find the probabilities P(A) and P(B) by finding the ratio of the number of favorable outcomes to the total (i. e., P(A) = #A #Ω and P(B) = #B #Ω ).

P(A) = 31/1296

P(B) = 125/1296

Step-by-step explanation:

Here, we want to find P(A) and P(B)

Let’s start with A

In a roll of a die, the probability that there will be a 5 is 1/6

The probability of not having a 5 is therefore 1-1/6 = 5/6

So out of four rolls , we want to find the probability of having at least two fives

This means there could be 2, 3 or 4 fives;

For two fives;

We have 2 fives and two non-fives

= (1/6 * 1/6 * 5/6 * 5/6) = 25/1296

For 3 fives;

we have 3 fives and 1 non five

= 1/6 * 1/6 * 1/6 * 5/6 = 5/1296

For 4 fives;

= 1/6 * 1/6 * 1/6 * 1/6 = 1/1296

So the probability of at least two fives is;

1/1296 + 5/1296 + 25/1296 = 31/1296

Event B is the probability that there is at most one five among the four rolls

That means there are 3 non fives and 1 five

P(B) = 1/6 * 5/6 * 5/6 * 5/6 = 125/1296

1. d.90

2. e. none of these (80)

3. a. 0

4. b. 6

5. e. none of these (10)

6. a. 5