Assume you roll a fair dice twice. Two rolls are independent and identically distributed, with probability of rolling a particular number being 1/6. So, for instance, the probability of rolling 5 and then 2 is P(5,2) = P(5) ⋅ P(2) = 1/6 ⋅ 1/6 = 1/36 Consider a random variable X equal to the sum of two numbers rolled. What is the probability of X being greater than 7?
Create the probability distribution for the random variable X (see other post), and sum probabilities of all outcomes greater than 7.
HOW? CAN YOU SHOW?
Did you read your other post?
http://www.jiskha.com/display.cgi?id=1370132243
Set up a table of outcomes and the corresponding probabilities to create the probability distribution.
For example, the lowest outcome is
X=2 (=1+1) with probability 1/36.
then
X=3 (1+2 or 2+1) with probability 2/36
...
X=12 (=6+6) with probability 1/36.
Find the mean using
E(X)=μ=Σ(X*P(X)) summed over the sample space.
and variance
σ²=Σ((X-μ)² *P(X)) again summed over sample space.
To find the probability of X being greater than 7, we need to calculate the probability of all possible outcomes where the sum of the two numbers rolled is greater than 7.
There are 36 possible outcomes when rolling two dice (6 options for the first roll and 6 options for the second roll, giving us 6*6 = 36 possible combinations).
We need to find the number of outcomes where the sum is greater than 7. Let's list them:
- Rolling a 6 on the first die and a 2, 3, 4, 5, or 6 on the second die. (5 possibilities)
- Rolling a 5 on the first die and a 3, 4, 5, or 6 on the second die. (4 possibilities)
- Rolling a 4 on the first die and a 4, 5, or 6 on the second die. (3 possibilities)
- Rolling a 3 on the first die and a 5 or 6 on the second die. (2 possibilities)
- Rolling a 2 on the first die and a 6 on the second die. (1 possibility)
So, there are a total of 5 + 4 + 3 + 2 + 1 = 15 outcomes where the sum of the two numbers rolled is greater than 7.
The probability of getting a sum greater than 7 is therefore P(X > 7) = 15/36 = 5/12.
To calculate this probability, we divided the number of outcomes where the sum is greater than 7 (15) by the total number of possible outcomes (36).
Therefore, the probability of X being greater than 7 is 5/12.