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# Bernoulli Distribution – Explanation & Examples

*The definition of the Bernoulli distribution is:*

**“The Bernoulli distribution is a discrete probability distribution that describes the probability of a random variable with only two outcomes.”**

*In this topic, we will discuss the Bernoulli distribution from the following aspects:*

- What is a Bernoulli distribution?
- When to use Bernoulli distribution?
- Bernoulli distribution formula.
- Practice questions.
- Answer key.

## 1. What is a Bernoulli distribution?

**The Bernoulli distribution** is a discrete probability distribution that describes the probability of a random variable with only two outcomes.

In the random process called a Bernoulli trial, the random variable can take one outcome, called a success, with a probability p, or take another outcome, called failure, with a probability q = 1-p.

The success outcome is denoted as 1 and the failure outcome is denoted as 0.

The Bernoulli distribution is a special case of the binomial distribution where a single trial is conducted and the binomial distribution is the sum of repeated Bernoulli trials.

**The Bernoulli distribution was named after the Swiss mathematician Jacob Bernoulli**.

### – Example 1

Tossing a coin can result in only two possible outcomes (head or tail). We call one of these outcomes (head) a success and the other (tail), a failure.

The probability of success (p) or head is 0.5 for a fair coin. The probability of failure (q) or tail = 1-p = 1-0.5 = 0.5.

*If we denote head as 1 and tail as 0, we can plot this Bernoulli distribution as follows:*

*We have two outcomes:*

- Tail or 0 with a probability of 0.5.
- Head or 1 with a probability of 0.5 also.

This is an **example of a probability mass function** where we have the probability for each outcome.

### – Example 2

We have an unfair coin where the probability of success (p) or head is 0.8 and the probability of failure (q) or tail = 1-p = 1-0.8 = 0.2.

*If we denote head as 1 and tail as 0, we can plot this Bernoulli distribution as follows:*

*We have two outcomes:*

- Tail or 0 with a probability of 0.2.
- Head or 1 with a probability of 0.8.

### – Example 3

The prevalence of a certain disease in the general population is 10%.

If we randomly select a person from this population, we can have only two possible outcomes (diseased or healthy person). We call one of these outcomes (diseased person) success and the other (healthy person), a failure.

The probability of success (p) or diseased person is 10% or 0.1. So, the probability of failure (q) or healthy person = 1-p = 1-0.1 = 0.9.

*If we denote diseased person as 1 and healthy person as 0, we can plot this Bernoulli distribution as follows:*

*We have two outcomes:*

- A healthy person or 0 with a probability of 0.9.
- A diseased person or 1 with a probability of 0.1.

### – Example 4

In the above example of disease prevalence of 10%, if We are interested in healthy persons and call the healthy person a success and the diseased person, a failure.

The probability of success (p) or healthy person is 90% or 0.9. So, the probability of failure (q) or diseased person = 1-p = 1-0.9 = 0.1.

*If we denote a healthy person as 1 and diseased person as 0, we can plot this Bernoulli distribution as follows:*

*We have two outcomes:*

- A healthy person or 1 with a probability of 0.9.
- A diseased person or 0 with a probability of 0.1.

## 2. When to use Bernoulli distribution?

*For a random variable to be described by the Bernoulli distribution:*

- The random variable can take only one of two possible outcomes. We call one of these outcomes a success and the other, a failure.
- The probability of success, denoted by p, is the same in every Bernoulli trial.
- The trials are independent, meaning that the outcome in one trial does not affect the outcome in other trials.

We can **determine the Bernoulli distribution** from the results of different Bernoulli trials.

### – Example 1

You are tossing a coin. The random variable equals to 1 if you get a head and 0 if you get a tail.

*You tossed the coin 100 times and get the following results:*

0 1 0 1 1 0 1 1 1 0 1 0 1 1 0 1 0 0 0 1 1 1 1 1 1 1 1 1 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 0 1 1 0 1 0 0 0 1 0 1 1 1 0 1 1 1 0 0 0 0 1 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 0 0 1 0 0 1.

What is the Bernoulli distribution for this coin?

You can use that data to estimate the probability mass function (or the probability distribution) for tossing this coin.

1. We construct a frequency table for each outcome.

Outcome | frequency |

0 | 53 |

1 | 47 |

2. Add another column for the probability of each outcome.

Probability = frequency/total number of data = frequency/100.

Outcome | frequency | probability |

0 | 53 | 0.53 |

1 | 47 | 0.47 |

The probabilities are >= 0 and sum to 1.

This is a likely fair coin where the probability of heads nearly equals the probability of tails = 0.5.

We do not get exactly 50 heads and 50 tails due to randomness in the process but we get a good approximation to the probability of the fair coin = 0.5.

3. Use the table to plot the Bernoulli distribution for that coin:

*We have two outcomes:*

- Head or 1 with a probability of 0.47.
- Tail or 0 with a probability of 0.53.

### – Example 2

You screened 50 individuals from a certain population for the presence of hypertension and get the following results:

ID | condition |

1 | normotensive |

2 | normotensive |

3 | normotensive |

4 | normotensive |

5 | normotensive |

6 | normotensive |

7 | normotensive |

8 | normotensive |

9 | normotensive |

10 | normotensive |

11 | hypertensive |

12 | normotensive |

13 | normotensive |

14 | normotensive |

15 | normotensive |

16 | normotensive |

17 | normotensive |

18 | normotensive |

19 | normotensive |

20 | hypertensive |

21 | normotensive |

22 | normotensive |

23 | normotensive |

24 | hypertensive |

25 | normotensive |

26 | normotensive |

27 | normotensive |

28 | normotensive |

29 | normotensive |

30 | normotensive |

31 | hypertensive |

32 | normotensive |

33 | normotensive |

34 | normotensive |

35 | normotensive |

36 | normotensive |

37 | normotensive |

38 | normotensive |

39 | normotensive |

40 | normotensive |

41 | normotensive |

42 | normotensive |

43 | normotensive |

44 | normotensive |

45 | normotensive |

46 | normotensive |

47 | normotensive |

48 | normotensive |

49 | normotensive |

50 | normotensive |

What is the estimated Bernoulli distribution for hypertension in this population?

1. We construct a frequency table for each outcome.

Outcome | frequency |

hypertensive | 4 |

normotensive | 46 |

2. Add another column for the probability of each outcome. As we are interested in hypertension, so we denote hypertensive persons as 1 and normotensive persons as 0.

Probability = frequency/total number of data = frequency/50.

outcome | frequency | probability |

1 | 4 | 0.08 |

0 | 46 | 0.92 |

The probabilities are >= 0 and sum to 1.

3. Use the table to plot the Bernoulli distribution for hypertension:

*We have two outcomes:*

- A hypertensive person or 1 with a probability of 0.08.
- A normotensive person or 0 with a probability of 0.92.

### – Example 3

You screened a 100 tablet-sample from two tablet producing machines in a certain factory. We denote 1 for rejected tablets and 0 for accepted tablets and get the following results:

Tablet | machine1 | machine2 |

1 | 0 | 0 |

2 | 0 | 0 |

3 | 0 | 0 |

4 | 0 | 1 |

5 | 0 | 0 |

6 | 0 | 0 |

7 | 0 | 1 |

8 | 0 | 0 |

9 | 0 | 0 |

10 | 0 | 0 |

11 | 1 | 1 |

12 | 0 | 0 |

13 | 0 | 0 |

14 | 0 | 1 |

15 | 0 | 0 |

16 | 0 | 0 |

17 | 0 | 0 |

18 | 0 | 1 |

19 | 0 | 0 |

20 | 1 | 0 |

21 | 0 | 0 |

22 | 0 | 0 |

23 | 0 | 0 |

24 | 1 | 0 |

25 | 0 | 0 |

26 | 0 | 1 |

27 | 0 | 0 |

28 | 0 | 0 |

29 | 0 | 0 |

30 | 0 | 0 |

31 | 1 | 0 |

32 | 0 | 0 |

33 | 0 | 0 |

34 | 0 | 0 |

35 | 0 | 0 |

36 | 0 | 0 |

37 | 0 | 0 |

38 | 0 | 0 |

39 | 0 | 1 |

40 | 0 | 0 |

41 | 0 | 0 |

42 | 0 | 0 |

43 | 0 | 0 |

44 | 0 | 0 |

45 | 0 | 0 |

46 | 0 | 0 |

47 | 0 | 0 |

48 | 0 | 0 |

49 | 0 | 0 |

50 | 0 | 0 |

51 | 0 | 0 |

52 | 0 | 0 |

53 | 0 | 0 |

54 | 0 | 0 |

55 | 0 | 0 |

56 | 0 | 0 |

57 | 0 | 0 |

58 | 0 | 0 |

59 | 0 | 0 |

60 | 0 | 0 |

61 | 0 | 0 |

62 | 0 | 0 |

63 | 0 | 0 |

64 | 0 | 0 |

65 | 0 | 0 |

66 | 0 | 0 |

67 | 0 | 0 |

68 | 0 | 0 |

69 | 0 | 0 |

70 | 0 | 0 |

71 | 0 | 0 |

72 | 0 | 0 |

73 | 0 | 0 |

74 | 0 | 0 |

75 | 0 | 0 |

76 | 0 | 0 |

77 | 0 | 0 |

78 | 0 | 0 |

79 | 0 | 0 |

80 | 0 | 0 |

81 | 0 | 0 |

82 | 0 | 0 |

83 | 0 | 0 |

84 | 0 | 0 |

85 | 0 | 0 |

86 | 0 | 0 |

87 | 1 | 0 |

88 | 0 | 0 |

89 | 0 | 1 |

90 | 0 | 1 |

91 | 0 | 0 |

92 | 0 | 0 |

93 | 0 | 1 |

94 | 0 | 0 |

95 | 0 | 1 |

96 | 0 | 0 |

97 | 0 | 0 |

98 | 0 | 0 |

99 | 0 | 0 |

100 | 0 | 0 |

What is the estimated Bernoulli distribution for rejections from each machine?

1. We construct a frequency table for each outcome.

Outcome | frequency_machine1 | frequency_machine2 |

0 | 95 | 89 |

1 | 5 | 11 |

2. Add another column for the probability of each outcome.

Probability = frequency/total number of data = frequency/100.

Outcome | frequency_machine1 | frequency_machine2 | probability_machine1 | probability_machine2 |

0 | 95 | 89 | 0.95 | 0.89 |

1 | 5 | 11 | 0.05 | 0.11 |

We see that the probability of rejected tablets from the second machine is 0.11 or 11% which is about double the probability from the first machine (0.05 or 5%).

3. Use the table to plot the Bernoulli distribution for these machines:

We see that the probability of rejections (outcome = 1) from the first machine is 0.05 or 5%, while the probability of rejections from the second machine is 0.11 or 11%.

## 4. Bernoulli distribution formula

If the random variable X takes only two outcomes, 0 or 1. The success outcome is denoted as 1 and the failure outcome is denoted as 0, and probability of success p, the probability of any outcome k is given by:

f(k,p)=p^k (1-p)^(1-k)

*where:*

f(k,p) is the probability of k outcome with a probability of success,p.

p is the probability of success and 1-p is the probability of failure.

*This probability mass function can take only two values as we have two outcomes only:*

When k =0, f(k,p)=p^k (1-p)^(1-k)=p^0 (1-p)^1 = 1-p.

When k =1, f(k,p)=p^k (1-p)^(1-k)=p^1 (1-p)^0 = p.

*So this probability mass function can be rewritten as:*

f(k,p)={■(p&”if ” k=1@1-p&”if ” k=0)┤

## 5. Practice questions

1. The following plot shows the Bernoulli distribution of diabetes in a certain population, where we denote the diabetic person as 1 and healthy person as 0:

What is the prevalence of diabetes in this population?

2. The following plot shows the Bernoulli distribution of survival from a certain pandemic in some population, where we denote the survived person as 1 and died person as 0:

If we know that this population is about 1 million persons. How many persons have survived this pandemic?

3. The following plot shows the Bernoulli distribution of survival from a certain pandemic in 2 populations, population1 and population2, where we denote the survived person as 1 and died person as 0:

Which population was more susceptible to this pandemic?

4. The following frequency table shows the number of hypertensive patients in a sample of 100 persons randomly selected from a certain population.

We denote hypertensive persons as 1 and normotensive persons as 0.

outcome | frequency |

0 | 94 |

1 | 6 |

What is the estimated Bernoulli distribution for hypertension in this population?

5. The following frequency table shows the number of failed students in a certain exam from 2 schools, school1 and school2.

We are interested in failed students, so we denote failed students as 1 and succeeded students as 0.

outcome | school1 | school2 |

0 | 65 | 86 |

1 | 5 | 14 |

What is the estimated Bernoulli distribution for failure in the 2 schools?

## 6. Answer key

1. We have two outcomes:

- A healthy person or 0 with a probability of 0.85.
- A diabetic person or 1 with a probability of 0.15.

The prevalence of diabetes in this population = probability of diabetic person = 0.15 or 15%.

2. We have two outcomes:

- A dead person or 0 with a probability of 0.2.
- A survived person or 1 with a probability of 0.8.

For 1 million persons, the number of survivors = 1,000,000 X 0.8 = 800,000 persons.

3. We have two outcomes:

- A dead person or 0.
- A survived person or 1 with a probability.

We see that population2 has more probability of dying (0.25) than population1 (0.15) so population2 was more susceptible.

4. We add another column for the probability of each outcome.

Probability = frequency/total number of data = frequency/100.

outcome | frequency | probability |

0 | 94 | 0.94 |

1 | 6 | 0.06 |

The probabilities are >= 0 and sum to 1.

Use the table to plot the Bernoulli distribution for hypertension:

So, this Bernoulli distribution can be written as:

f(k,p)={■(0.06&”if ” [email protected]&”if ” k=0)┤

5. We note that school1 has 70 students, while school2 has 100 students.

Add another column for the probability of each outcome.

For school1, Probability = frequency/total number of data = frequency/70.

For school2, Probability = frequency/total number of data = frequency/100.

outcome | school1 | school2 | probability_school1 | probability_school2 |

0 | 65 | 86 | 0.93 | 0.86 |

1 | 5 | 14 | 0.07 | 0.14 |

We see that the probability of failure for students from school1 is 0.07 or 7% which is half the probability from school2 (0.14 or 14%).

Use the table to plot the Bernoulli distribution for the 2 schools:

*For school1, the Bernoulli distribution can be written as:*

f(k,p)={■(0.07&”if ” [email protected]&”if ” k=0)┤

*and for school2, the Bernoulli distribution can be written as:*

f(k,p)={■(0.14&”if ” [email protected]&”if ” k=0)┤