Aim: What is the importance of probability?

What is the language of Probability? “Random” is a description of a kind of order that emerges in the long run We call a phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a larger number of repetitions The probability of any outcome of random phenomenon is the proportion of times the outcome would occur in a very long series of repetitions – In probability we assume fair even though not everything is really fair – Probability describes what happens in very many trials, and we must actually observe many trials to pin down a probability.

What is probability theory? Probability theory is the branch of mathematics that describes random behavior. Mathematical probability is an idealization based on imagining what would happen in an indefinitely long series of trials.

Exploring Randomness You must have a long series of independent trials. – That is, the outcome of one trial must not influence the outcome of any other. The idea of probability is empirical – Simulations start with given probabilities and imitate random behavior, but we can estimate a real-world probability only by actually observing many trials. Simulations are very useful because we need long runs of trials. – In situations such as coin tossing, the proportion of an outcome often requires several hundred trials to settle down to the probability of that outcome. The kinds of physical random devices suggested in the exercises are too slow for this. Short runs give only rough estimates of a probability.

The Uses of Probability Probability theory originated in the study of games of chance. – Tossing dice, dealing shuffled cards, and spinning a roulette wheel are examples of deliberate randomization. Probability is used in astronomy, math, surveying, economics, genetics, biology etc. – Although we are interested in probability because of its usefulness in statistics, the mathematics of chance is important in many fields of study.

Pop Quiz 1.When is a phenomenon random? 2.What is the probability of an event? 3. What is the probability theorem? 4.Describe three bullet-points of exploring randomness? 5.How did the probability theory develop? 6.What are examples of deliberate randomization?

Answers to Pop Quiz 1.A phenomenon random if individual outcomes are uncertain but there is nonetheless a regular distribution of outcomes in a larger number of repetitions 2.The probability of any outcome is the proportion of times the outcome would occur in a very long series of repetitions 3.Probability theory is the branch of mathematics that describes random behavior. 4.(1) You must have a long series of independent trials. (2) The idea of probability is empirical (3) Simulations are very useful because we need long runs of trials. 5.Probability theory originated in the study of games of chance. 6.Tossing dice, dealing shuffled cards, and spinning a roulette wheel are examples of deliberate randomization.

Class Work 1.Use Table B. We can use the random digits in Table B in the back of the text to simulate tossing a fair coin. Start at line 109 and read the numbers from left to right. If the number is 0, 1, 2, 3, or 4, you will say that the coin toss resulted in a head; if the number is a 5, 6, 7, 8, or 9, the outcome is tails. Use the first 20 random digits on line 109 to simulate 20 tosses of a fair coin. What is the actual proportion of heads in your simulated sample? Explain why you did not get exactly 10 heads. 2.You go to the doctor and she prescribes a medicine for an eye infection that you have. Suppose that the probability of a serious side effect from the medicine is Explain in simple terms what this number means.