2 Recap

Probability is a mathematical tool used to study randomness. It deals with the chance (the likelihood) of an event occurring. The theory of probability began with the study of games of chance such as poker. Predictions take the form of probabilities. To predict the likelihood of an earthquake, of rain, or whether you will get an A in this course, we use probabilities. Doctors use probability to determine the chance of a vaccination causing the disease the vaccination is supposed to prevent. A stockbroker uses probability to determine the rate of return on a client’s investments. You might use probability to decide to buy a lottery ticket or not. In your study of statistics, you will use the power of mathematics through probability calculations to analyze and interpret your data.

2.1 Learning objectives

Having read this chapter, you should be able to:

Describe the sample space for a selected random experiment. Describe the law of large numbers. Describe the difference between a probability and a conditional probability Describe the relationship between z-scores and the standard unit normal table (z-table) Probability is a tough topic for everyone, but the tools it gives us are incredibly powerful and enable us to do amazing things with data analysis. They are the heart of how inferential statistics work.

2.2 Exercises

  1. In your own words, what is probability?
  2. There is a bag with 5 red blocks, 2 yellow blocks, and 4 blue blocks. If you reach in and grab one block without looking, what is the probability it is red?
  3. Under a normal distribution, which of the following is more likely? (Note: this question can be answered without any calculations if you draw out the distributions and shade properly)
    • Getting a z-score greater than z = 2.75
    • Getting a z-score less than z = -1.50
  4. The heights of women in the United States are normally distributed with a mean of 63.7 inches and a standard deviation of 2.7 inches. If you randomly select a woman in the United States, what is the probability that she will be between 65 and 67 inches tall?
  5. The heights of men in the United States are normally distributed with a mean of 69.1 inches and a standard deviation of 2.9 inches. 6. What proportion of men are taller than 6 feet (72 inches)?
  6. You know you need to score at least 82 points on the final exam to pass your class. After the final, you find out that the average score on the exam was 78 with a standard deviation of 7. How likely is it that you pass the class?
  7. What proportion of the area under the normal curve is greater than z = 1.65?
  8. Find the z-score that bounds 25% of the lower tail of the distribution.
  9. Find the z-score that bounds the top 9% of the distribution.
  10. In a distribution with a mean of 70 and standard deviation of 12, what proportion of scores are lower than 55?