2.2.1.2 Binomial Distribution
Binomial Distribution is a distribution in which every trial has only two possible outcomes. The distribution has some key characteristics:
- There are two potential outcomes per trial
- The number of trials (denoted by
) is fixed - The probability of success (denoted by
) is the same for all the trials - Each trial is independent
The term “Binomial” inherently signifies the presence of two terms. For instance, it can represent outcomes like heads or tails, true or false, or yes or no. In the context of binomial distribution, one outcome is typically labeled as 'success,' while the other is termed 'failure'.
Binomial distribution can be calculated through the following formula:
Example 2.3. A study shows that 20% of pregnant women suffer from diabetes during pregnancy, A random sample of 10 women is taken. Find the probability of the following scenarios:
- All 10 women are diabetic
- No women are diabetic
- Exactly 4 women are diabetic
- At least 4 women are diabetic
As the problem has two outcomes (Diabetic or non-diabetic women), it can be
considered a binomial distribution. Here, the values of and are the same for all the scenarios:
- (number of trials) = 10
- (probability of diabetic women) = 0.2
Scenario 1: All 10 women are diabetic
Here, 
(number of successes i.e., diabetic women for whom we want to calculate
the probability) = 10
Substituting the values of ,, and in the binomial distribution formula yields:
