The bulbs manufactured by a company gave a mean life of 3000 hours with standard

deviation of 400 hours. If a bulb is selected at random, what is the probability it will
have a mean life less than 2000 hours?
Question:
1) Calculate the probability.
2) In what situation does one need probability theory?

CASE STUDY: 1

The bulbs manufactured by a company gave a mean life of 3000 hours with standard
deviation of 400 hours. If a bulb is selected at random, what is the probability it will
have a mean life less than 2000 hours?
Question:
1) Calculate the probability.
2) In what situation does one need probability theory?
3) Define the concept of sample space, sample points and events in context of
probability theory.
4) What is the difference between objective and subjective probability?

Step 1: Calculate the z-score.

To calculate the probability, we first need to calculate the z-score using the formula:

z = (x - μ) / σ

where:
x = mean life we want to calculate the probability for (2000 hours)
μ = mean life of the bulbs (3000 hours)
σ = standard deviation of the bulbs (400 hours)

Plugging in the values, we get:

z = (2000 - 3000) / 400
z = -1

Step 2: Find the probability using the z-table.

Using the z-table, we can find the probability corresponding to the z-score. Since we want to find the probability of a mean life less than 2000 hours, we need to find the area under the curve to the left of the z-score (-1).

Looking up the z-score of -1 in the z-table, we find that the corresponding probability is 0.1587.

Step 3: Calculate the probability.

The probability of a bulb having a mean life less than 2000 hours is 0.1587, or 15.87%.

Answer to question 1: The probability is 0.1587 or 15.87%.

Answer to question 2: Probability theory is used in situations where there is uncertainty or randomness involved. It helps to predict and analyze the likelihood of certain events or outcomes occurring based on the available information. Probability theory is used widely in various fields such as mathematics, statistics, economics, finance, and physics, among others.

To calculate the probability of a random bulb having a mean life less than 2000 hours, we can use the concept of standardizing a normal distribution.

1) Calculate the probability:
First, let's standardize the value of 2000 hours using the formula:
z = (x - μ) / σ
where z is the standardized score, x is the value, μ is the mean, and σ is the standard deviation.

In this case, the mean (μ) is 3000 hours and the standard deviation (σ) is 400 hours. Therefore, plugging in the values:
z = (2000 - 3000) / 400
z = -1

Next, we need to find the probability of getting a z-score less than -1. We can refer to a standard normal distribution table or use statistical software to find this probability.

Looking up the z-score of -1 in the standard normal distribution table, we find that the corresponding cumulative probability is approximately 0.1587. Therefore, the probability of a random bulb having a mean life less than 2000 hours is approximately 0.1587 or 15.87%.

2) In what situation does one need probability theory?
Probability theory is used in a wide range of situations where uncertainty or randomness is involved. Here are a few examples:

a) Gambling: When playing games of chance, probability theory helps in determining the likelihood of winning and making strategic decisions.

b) Risk analysis: In fields like insurance and finance, probability theory is used to assess and manage risks. It helps in estimating the probability of various outcomes and determining appropriate strategies.

c) Decision-making: Probability theory provides a framework for making rational decisions under uncertainty. It helps in considering the likelihood of different outcomes and weighing potential risks and rewards.

d) Statistics: Probability theory is the foundation of statistical analysis. It is used to design experiments, analyze data, and make inferences about populations based on samples.

e) Machine learning and AI: Probability theory plays a crucial role in designing and training machine learning models. It helps in modeling uncertainty, estimating probabilities, and making predictions based on data.

Overall, probability theory is essential in various fields where understanding and quantifying uncertainty is necessary for making informed decisions and predictions.