Probability of At Least One

Probability of at least one normally describes the chance that an event will take place at least once within a sequence of chances or trials. This concept is widely applicable in various fields, from predicting the chances of at least one rainy day in a week to assessing the risk of at least one machine failure in a manufacturing process. In this article, we will solve different problems related to the probability of at least one event.

Defining Probability of At Least One

Probability of at least one event occurring is calculated by determining the complement of the event where none of the specified events happen. This approach leverages the complementary rule in probability, which states that the probability of the complement of an event is equal to one minus the probability of the event itself.

Mathematical Representation of Probability of At Least One

If we consider several independent events A₁, A₂,…, A_n​, the probability of at least one of these events occurring is given by:

P(at least one A_i​) = 1 - P(none of the A_i​)

To find P(none of the A_i), we multiply the probabilities of each event not occurring:

Here, represents the complement of event A_i​, i.e., the event that A_i​ does not occur.

Application of Probability of At Least One

The probability of at least one event occurring is a versatile concept with a wide range of applications across various fields. This principle helps in risk assessment, decision-making, and strategic planning. Let's explore some detailed applications of this concept in different domains.

Quality Control in Manufacturing

Example: A factory produces widgets, and it is known that the defect rate is 2%. If a quality control team tests 50 widgets, the probability of at least one defective widget can be calculated to decide if the batch needs further inspection or rejection.

Network Security

Example: Suppose a company faces 100 cyber-attack attempts daily, each with a 0.1% chance of success. Calculating the probability of at least one successful attack informs the company about the necessary security measures and resources to allocate for protection.

Medical Screening

Example: In a population screening for a rare disease with a prevalence of 1 in 1,000, testing 200 people provides a probability of at least one positive result. This information guides healthcare providers in resource allocation and treatment strategies.

Environmental Studies

Example: An area has an annual flood risk of 5%. Over a 10-year period, the probability of experiencing at least one flood helps in planning infrastructure projects, insurance policies, and emergency response strategies.

Marketing Campaigns

Example: A company sends out 1,000 promotional emails with an expected conversion rate of 1%. Calculating the probability of at least one conversion helps in assessing the effectiveness of the campaign and making data-driven decisions for future marketing efforts.

Read More:

• Bayes’ Theorem
• Conditional Probability
• Bernoulli Trials and Binomial Distribution
• Probability Distribution Function

Examples of Probability of At Least One

Example 1: If you roll a fair six-sided die 3 times. What is the probability of rolling at least one 6?

Solution:

Probability of not rolling a 6 in one roll = 5/6

Probability of not rolling a 6 in all three rolls = (5/6)³ = 125/216

Probability of rolling at least one 6 = 1 - 125/216 = 91/216

Example 2: From a standard deck of 52 cards, what is the probability of drawing at least one ace when drawing 5 cards?

Solution:

Total ways to draw 5 cards from 52 cards = C(52, 5) = 2,598,960

Ways to draw 5 cards with no aces: C(48, 5) = 1,712,304 (since there are 48 non-ace cards)

Probability of no aces = P(no aces) = C(48, 5) / C(52, 5) = 1,712,304 / 2,598,960

Calculating the Probability of at Least One Ace

P(at least one ace) = 1 - P(no aces)

P(at least one ace) = 1 - (1,712,304 / 2,598,960) ≈ 0.3457

Example 3: A coin is tossed 4 times. What is the probability of getting at least one head?

Solution:

P(no heads) = (1/2)⁴ = 1/16

P(at least one head) = 1 - 1/16 = 15/16

Example 4: A box contains 5 red and 3 blue marbles. Two marbles are drawn randomly without replacement. What is the probability of getting at least one red marble?

Solution:

P(no red marbles) = (3/8) × (2/7) = 3/28

P(at least one red marble) = 1 - 3/28 = 25/28

Example 5: A family has 3 children. What is the probability that at least one child is a girl

Solution:

P(no girls) = (1/2)³ = 1/8

P(at least one girl) = 1 - 1/8 = 7/8