6.7 Relative Frequency

Example Suppose you throw a dart 100 times at the dart board below. What is the probability of hitting the 1? Is it 1 5 ? 5 4 3 1

Relative Frequency For some problems, equally likely probability is not reasonable. If it is possible to preform the experiment many times, or to observe the results of experimentation, we can use the concept of relative frequency as a means of assigning probabilities to simple events.

# Times Achieved in n Throws Example Suppose you throw a dart 1000 times at the dart board below. Summarizing and recording your scores after 10, 100, 200, 500, and 1000 throws gives the results shown in table form. # Times Achieved in n Throws Score n=10 n=100 n=200 n=500 n=1000 1 6 11 24 55 5 14 37 74 3 4 41 86 206 410 46 82 215 426 2 7 15 35 Total # Throws 10 100 200 500 1000 5 4 3 1

# Times Achieved in n Throws Example Continued Find the following probabilities: Probability of getting a 0 score After 10 throws=1/10 = 0.100 After 100 throws = 6/100 = 0.060 After 200 throws = 11/200 = 0.055 After 500= 24/500 = 0.054 After 1000= 55/1000 = 0.055 As, the throws increase, the relative frequency stabilizes. Relative freq. is an approximation of probabilities. # Times Achieved in n Throws Score n=10 n=100 n=200 n=500 n=1000 1 6 11 24 55 5 14 37 74 3 4 41 86 206 410 46 82 215 426 2 7 15 35 Total # Throws 10 100 200 500 1000

Example Medical records of 140 deceased males revealed the following: Find the probability that a male randomly chosen from this group died of (a) heart disease; (b) heart disease or stroke; (c) cancer; (d) Find the probability that one who lived to age 40 died of a stroke. Age at Time of Death Cause of Death Under 40 40-60 Over 60 Heart Disease 6 14 26 Cancer 4 9 12 Stroke 2 7 Pneumonia 1 Diabetes Tuberculosis Other 15 11 20

Solutions 6+14+26=46→ 46 140 =0.329 46+13=59→ 59 140 =0.421 25 140 =0.179 4+7=11→ 11 140 =0.0982

Vocab Probability obtained using relative frequency is called objective or empirical probability because values are based on experimental results. Sometimes an equally likely approach is unwarranted, but the experiment cannot be repeated in order to use relative frequency.

Vocab Subjective or personal probability can be helpful when using probability to make a personal decision. For example: Charles Novak, a college junior, hopes to borrow $4000 towards senior-year expenses. If the bank officer knows Charles and has lent him money before, he would overlook the statistics that 238 out of 1200 junior-year borrowers failed to repay their loans.

Homework Page 41 #173-195