Question

How do you determine the probability distribution of $P\left( x>5 \right)$ ? The table :

x	4	5	6	7	8
P(x)	0.1	0.1	0.2	0.1	0.5

Answer 1

In this question, we have to find the probability from a distribution table. As we know, a probability distribution is a statistical measure that consists of all the possible values of a random variable. Thus, we will use the probability formula and the basic mathematical rule to get the solution. First, we will write the meaning of $P\left( x>5 \right)$ , which is equal to the sum of probabilities when x is equal to 6, 7, and 8. Then, we will substitute the value of the probability given in the table in the formula, to get the required solution for the problem.

Complete step-by-step solution:
According to the problem, we have to find the value of a probability.
Thus, we will use the probability rule and the basic mathematical rule to get the solution.
The probability table given to us is

x	4	5	6	7	8
P(x)	0.1	0.1	0.2	0.1	0.5

The table (1) signifies that when x=4, its probability is equal to 0.1, when x=5, its probability is equal to 0.2, similarly for x=6, 7, and 8, that is
$P(4)=0.1,$ $P(5)=0.1,$ $P(6)=0.2,$ $P(7)=0.1,$ and $P(8)=0.5$ --------- (2)
Now, we have to find the value of $P\left( x>5 \right)$, we know when x is greater than 5, it implies x will take values as 6, 7, 8, and so on. Since, in the table we have till 8 values. Therefore, the probability of x greater than 5 is the summation of probability when x=6, 7, and 8, that is
$P\left( x>5 \right)=P\left( x=6 \right)+P\left( x=7 \right)+P\left( x=8 \right)$
Now, we will substitute the value of equation (2) in the above equation, we get
$P\left( x>5 \right)=0.2+0.1+0.5$
On further solving the right-hand side of the above equation, we get
$P\left( x>5 \right)=0.8$
Therefore, the value of $P\left( x>5 \right)$ from the table is equal to 0.8

Note: While solving this problem, always remember the meaning of $P\left( x>5 \right)$ , do not add the probability when x=5. One of the alternative methods to solve this problem is using the formula $P\left( x>5 \right)=1-P\left( x\le 5 \right)$.