Question

Do these situations involve Bernoulli trials? Explain. a) We roll 50 dice to find the distribution of the number of spots on the faces. b) How likely is it that in a group of 120 the majority may have Type A blood, given that Type A is found in \(43 \%\) of the population? c) We deal 7 cards from a deck and get all hearts. How likely is that? d) We wish to predict the outcome of a vote on the school budget, and poll 500 of the 3000 likely voters to see how many favor the proposed budget. e) A company realizes that about \(10 \%\) of its packages are not being sealed properly. In a case of 24, is it likely that more than 3 are unsealed?

Short answer

Scenarios b, d, and e involve Bernoulli trials.

Step-by-step solution

Understand Bernoulli Trials

First, recognize that a Bernoulli trial is an experiment with exactly two possible outcomes, commonly labeled as "success" and "failure". Each trial is independent, and the probability of success remains constant across trials.

Evaluate Scenario a

In scenario (a), rolling 50 dice does not qualify as Bernoulli trials. This is because a die roll can result in any one of six outcomes (1 to 6), which does not align with the Bernoulli trial requirement of having only two outcomes.

Evaluate Scenario b

In scenario (b), since each person in the group can either have Type A blood (success) or not have Type A blood (failure), this scenario fits the Bernoulli trials criteria. The trials are independent and each has the same probability (43%) of succeeding.

Evaluate Scenario c

In scenario (c), when dealing cards from a deck without replacement, the probability of drawing a heart changes with each draw, making the trials dependent. Thus, this does not fit the conditions for Bernoulli trials.

Evaluate Scenario d

Scenario (d) involves polling voters, typically independent and with two outcomes—favoring or not favoring the budget. If the poll is random, each person has a fixed probability of favoring the budget, fulfilling the Bernoulli criteria.

Evaluate Scenario e

In scenario (e), each package being properly sealed or not is an independent event with a constant 10% probability of failure. This meets the requirements for Bernoulli trials.

Key concepts

Probability

Probability is a fundamental concept in mathematics, and it describes how likely an event is to occur. It is a numerical expression of chance and is expressed as a number between 0 and 1. The closer the probability is to 1, the more likely the event is to occur. Meanwhile, a probability of 0 means the event is impossible.

When calculating the probability of an event, you consider the number of successful outcomes divided by the total number of possible outcomes. For instance, when flipping a fair coin, there are two possible outcomes, heads or tails. The probability of getting heads is \\[ P(\text{heads}) = \frac{1}{2} = 0.5 \\]

To apply probability to a Bernoulli trial, you'll assign a probability to one of the outcomes, often referred to as "success." This process depends on having two outcomes where the success rate remains constant across multiple trials.

Independent Events

Independent events are a key component of Bernoulli trials. When events are independent, the occurrence of one event does not influence the occurrence of another. In simpler terms, each event has its own separate probability that remains unchanged regardless of any other event.

For example, with a fair die, each roll is independent. The probability of rolling a 4 remains 1/6, regardless of previous rolls. With a Bernoulli trial, this independence is crucial, as it ensures that each trial's probability of success fixed (e.g., rolling a die to get a 4) remains unchanged.

This is significant in scenarios like example (e) where a company evaluates if packages are sealed or not. Here, each package represents an independent trial, and the probability (10%) is constant across packages. This independence helps in accurately calculating overall probabilities by multiplying the single probabilities of each independent event.

Success and Failure Outcomes

In the context of Bernoulli trials, the concepts of success and failure are central. Each trial can result in one of two possible outcomes, and these are typically simplified into 'success' or 'failure'. Here, 'success' does not necessarily mean a positive result; it is simply the outcome of interest for the experiment.

For example, if you are interested in how many people in a group have Type A blood, having Type A blood would be defined as 'success.' Conversely, not having Type A blood would be a 'failure.'

In Bernoulli trials, consistency in these definitions across trials is important. For the company in example (e) trying to predict unsealed packages, having more than three unsealed is significant, defining unsealing as 'success' for calculation purposes. Similarly, in voting preferences, 'success' might revolve around favor in polling or predicting outcomes. By consistently applying these definitions, the probability of these outcomes can be examined across repeated trials.