Question

What is the sum of the probabilities in a probability distribution?

Answer 1

Hint: To find the sum of the probabilities in a probability distribution, we have to consider an example. Let us consider tossing a coin. We have to find the probabilities of the outcomes in this experiment and sum these probabilities.

Complete step-by-step solution:
We have to find the sum of the probabilities in a probability distribution. Let us see what a probability distribution is. Probability distribution holds the possible outcomes of any random event. It is also defined based on the underlying sample space as a set of possible outcomes of any random experiment.
Let us see the rules that probability distribution holds on to. The first rule is that each probability in the distribution must be of a value between 0 and 1. The second rule is that the sum of all the probabilities in the distribution must be equal to 1.
Let us consider an event of tossing a coin. We know that when a coin is tossed, there can only be 2 outcomes. These are the occurrence of head and tail. Hence, we can write the sample space as
$S=\{H,T\}$
Let us find the probability of occurring in a head. We know that probability of an event E, is given by
$P\left(E\right)={\displaystyle \frac{\text{Number of favourable outcomes}}{\text{Total number of outcomes}}}$
In this example, we can see that number of favourable outcomes, that is, head, occurs only once. The total number of outcomes is the number of elements in the sample space. Here, this value is 2. Hence, probability of occurring a head is
$P\left(H\right)={\displaystyle \frac{1}{2}}$
Now, let us find the probability of occurring a tail. We can see that number of favourable outcomes, that is, tail, occurs only once. Hence, probability of occurring a tail is
$P\left(T\right)={\displaystyle \frac{1}{2}}$
We can see from the above probabilities that each probability in the distribution is valued between 0 and 1. Hence, the rule one is true.
Now, let us see the sum of probabilities.
$P\left(H\right)+P\left(T\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{2}}={\displaystyle \frac{2}{2}}=1$
Hence, the sum of the probabilities in a probability distribution is always 1.

Note: Students must know what probability is and how to find it. Probability is a measure of uncertainty of various phenomena. They can also check whether the sum of probabilities is always 1 using other examples also, like rolling of a dice.