A pair of dice is cast. What is the probability that the sum of the two number landing uppermost is less than 5, if it is known that the sum of the numbers falling uppermost is less than 7?
To find the probability that the sum of the two numbers on the dice is less than 5, given that the sum is less than 7, we need to calculate the probability of each possible outcome and then divide it by the total number of favorable outcomes.
There are a total of 36 possible outcomes when two dice are rolled (6 possible outcomes for the first dice and 6 possible outcomes for the second dice).
Let's list all the possible outcomes where the sum is less than 7:
1+1 = 2
1+2 = 3
1+3 = 4
1+4 = 5
2+1 = 3
2+2 = 4
2+3 = 5
3+1 = 4
3+2 = 5
4+1 = 5
Out of these 10 possible outcomes, only 4 of them have a sum that is less than 5:
1+1 = 2
1+2 = 3
1+3 = 4
2+1 = 3
Therefore, there are 4 favorable outcomes out of 36 possible outcomes.
To calculate the probability, divide the number of favorable outcomes by the total number of possible outcomes:
Probability = Favorable outcomes / Total outcomes
Probability = 4 / 36
Simplifying the fraction, we get:
Probability = 1 / 9
So, the probability that the sum of the two numbers landing uppermost is less than 5, given that the sum is less than 7, is 1/9.
To find the probability that the sum of the two numbers landing uppermost on a pair of dice is less than 5, given that the sum is less than 7, we need to determine the sample space and the favorable outcomes.
Let's first consider the sample space, which represents all possible outcomes. Each die has 6 faces, numbered from 1 to 6, so there are 6 possible outcomes for each die. Since we are casting a pair of dice, we multiply the number of outcomes by itself, resulting in a total of 6 * 6 = 36 possible outcomes.
Next, we need to determine the favorable outcomes that satisfy the condition that the sum of the numbers landing uppermost is less than 5, given that it is less than 7. We can list all the possible outcomes:
1 + 1 = 2
1 + 2 = 3
1 + 3 = 4
1 + 4 = 5
2 + 1 = 3
2 + 2 = 4
2 + 3 = 5
3 + 1 = 4
3 + 2 = 5
4 + 1 = 5
There are a total of 10 favorable outcomes.
Finally, we can find the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
Probability = 10 / 36
Probability = 5 / 18
Therefore, the probability that the sum of the two numbers landing uppermost is less than 5, given that it is less than 7, is 5/18.