if a pair of fair dice is rolled, what is the probability that we roll a total of five if we are given that the total is less than nine?
total less than nine
lets look at the ways to get nine or more
4,5; 4,6 four ways to do that
5,5; 5,6 four ways to do that
6,6 one way to do that.
Now, ways to get less than 9=
ways to throw two die, minus way to get nine or more= 36-9above=29 ways check that.
Now, if we roll a five, we have to
4,1 two ways to do that
3,2 two ways to do that.
Pr(5, given nine or less)=4/29
check all this, I have been up all night.
4/30
To calculate the probability of rolling a total of five given that the total is less than nine, we need to determine the favorable outcomes and the total number of possible outcomes.
Step 1: Determine the favorable outcomes:
We want to roll a total of five, so we need to find all the possible combinations of two dice that add up to five. The combinations are: (1, 4), (2, 3), and (3, 2). So, there are three favorable outcomes.
Step 2: Determine the total number of possible outcomes:
Since we are given that the total is less than nine, we need to find all the possible combinations where the sum of the two dice is less than nine. We can start by finding the total number of combinations for each possible sum:
For a sum of two, the combinations are: (1, 1)
For a sum of three, the combinations are: (1, 2), (2, 1), (1, 2)
For a sum of four, the combinations are: (1, 3), (2, 2), (3, 1)
For a sum of five, the combinations are: (1, 4), (2, 3), (3, 2), (4, 1)
For a sum of six, the combinations are: (1, 5), (2, 4), (3, 3), (4, 2), (5, 1)
For a sum of seven, the combinations are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)
For a sum of eight, the combinations are: (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)
For a sum of nine, the combinations are: (3, 6), (4, 5), (5, 4), (6, 3)
For a sum of ten, the combinations are: (4, 6), (5, 5), (6, 4)
For a sum of eleven, the combinations are: (5, 6), (6, 5)
For a sum of twelve, the combinations are: (6, 6)
Adding up all these combinations, we find that there are 36 possible outcomes.
Step 3: Calculate the probability:
To calculate the probability, we divide the favorable outcomes by the total number of possible outcomes:
Probability = (number of favorable outcomes) / (total number of possible outcomes)
Probability = 3 / 36
Probability = 1/12
Therefore, the probability of rolling a total of five, given that the total is less than nine, is 1/12.
To find the probability of rolling a total of five given that the total is less than nine, we need to consider the possible outcomes and calculate the probability.
Step 1: Determine the favorable outcomes
When two fair dice are rolled, there are 36 equally likely outcomes (6 possible outcomes on the first die multiplied by 6 possible outcomes on the second die).
Out of these 36 outcomes, we need to identify the favorable outcomes, which are the outcomes where the total is five and the total is less than nine.
The possible outcomes that satisfy both conditions are: (1, 4), (2, 3), (3, 2), and (4, 1).
Step 2: Calculate the probability
The probability is calculated by dividing the number of favorable outcomes by the total number of outcomes.
The favorable outcomes are 4, and the total number of outcomes is 36.
Therefore, the probability of rolling a total of five given that the total is less than nine is 4/36, which simplifies to 1/9.
So the probability is 1/9.