5: Modeling and Finding Probabilities

5.1

Experimental Probability

Read the textbook

• Key Idea 1: The types of questions we will consider...



Discussion

If a basketball player makes 60% of her shots, what is the chance that she will get four baskets in her next six tries? What is the probability that a family of four children will have children of both sexes? If light bulbs last on average six months, what is the chance a particular light bulb will last nine months? If I make an airline reservation, what is the chance that, because of overbooking, I will not be able to get a seat on the airplane? A lot of 50 VCRs contains 10 defectives. If one randomly chooses 10 of the 50 VCRs, what is the probability that none are defective?




• Key Idea 2: It was in the Middle Ages when gamblers (and mathematicians and philosophers) began to think about probability in a more systematic way-with the obvious hope of improving their warnings!...



Discussion

The serious study of probability apparently began in the seventeenth century when the French nobleman Chevalier de Mere approached the famous mathematician Blaise Pascal with questions about betting strategies in games of chance. Pascal took up challenge and, together with another famous mathematician, Pierre de Fermat did the groundwork for the formal rules of probability theory.




• Key Idea 3: In this book we use an experimental, or relative frequency, interpretation of probability...



Discussion

This interpretation simply says that an observed "experimental probability" resulting from many trials (for example, many tosses of a coin) provides a good estimate of the unknown theoretical probability. Indeed, the experimental probability that would result if we kept doing such trials forever ("in the limit") is, in fact, the theoretical probability under this relative frequency interpretation. Here is an example. In each of these examples, The ratio obtained in this manner is an experimental probability, based on the relative frequency (that is, the frequency or count relative to the total number of trials) of the desired event occurring.

• Key Idea 4: Probabilities must be between 0 and 1...
  

• Key Idea 5: The more events or trials an experimental probability is based upon, the more we can trust it as an estimate of the true, or theoretical probability...



Discussion

J.E. Kerrick, while a prisoner during World War II, tossed a coin 10,000 times and noted the result after 10, tosses, 20 tosses, 30 tosses.... Note how the cumulative proportion of heads varies less and less over time, and settles down to a value pretty close to 0.5. Number of tosses Number of heads Relative frequency (experimental probability) 10 6  0.600 20 6  0.300 30 10 0.333 40 14 0.350 50 22 0.440 100 48 0.480 150 66 0.449 200 88 0.440 300 136 0.453 400 182 0.455 500 235  0.470 600 286  0.477 700 336  0.480 800 384  0.480 900 431  0.479 1000 486  0.486 2000 995  0.498 3000 1498 0.501 4000 2001 0.500 5000 2502 0.500 6000 3009 0.502 7000 3547 0.507 8000 4059 0.507 9000 4541 0.504 10000 5056  0.506




• Key Idea 6: The fact that experimental probabilities stabilize as the number of trials increases is one of the fundamental laws of science...



Discussion

This is sometimes called the law of statistical regularity. We cannot predict one coin toss, but we can predict quite accurately that the proportion of heads in 10,000 tosses will be close to the theoretical probability.

Exercises

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