Question

Find the value of k in the given probability distribution:

${{x}_{i}}$	2	3	4
${{p}_{i}}$	0.2	k	0.3

Answer 1

Hint: Given it the probability distribution. Probability distribution contains all possibilities and their probabilities respectively. So, from the question find the probabilities and add them together. By equating them to one you can find the variable remaining.

Complete step-by-step solution -
Probability distribution: A probability distribution is a representation which contains the information of likelihood of obtaining the possible values that a random variable can assume. In other words we can say the variable varies based on the underlying probability distribution.
It contains all the possibilities. So, if we sum them together we must get a total probability which is always 1. For an experiment the total probability is always 1.
Probability can be in the form of a mathematical table, mathematical relation or a mathematical function. ${{x}_{i}}$’s represent the possibility of an experiment and ${{p}_{i}}$’s represent the probability of that possibility respectively.
By using the possibility of given experiment, we get:
${x_1}=2,\,\,\,\,{{x}_{2}}=3,\,\,\,\,\,\,\,{{x}_{3}}=4$
By listing the probabilities of each possibilities, we get:
\[{{p}_{1}}=0.2,\,\,\,\,{{p}_{2}}=k,\,\,\,\,\,\,\,{{p}_{3}}=0.3\]
Now, by the definition we know that the sum of probability must be equal to 1.
Let us assume:
\[{{p}_{1}}=a,\,\,\,\,{{p}_{2}}=b,\,\,\,\,\,\,\,{{p}_{3}}=c\]
By substituting the above values into the condition, we get:
$a+b+c=1$
We know that values of a, b, c to be 0.2, k, 0.3.
By substituting these into above equation, we get:
$0.2+k+0.3=1$
By simplifying the above equation, we can write it as:
$k+0.5=1$
By subtracting 0.5 on both sides of equation, we get:
$k=1-0.5$
By simplifying the above equation, we get:
$k=0.5$
Therefore, the value of k from the given distribution is 0.5.

Note: Be careful while taking probability, don't confuse between possibility and probability. Possibility is outcome, probability is chance of that outcome.