To Lecture Notes

ISP 121 -- 5/12/08

Review Questions

1. What does the correlation between two variables mean?

2. How high must a correlation be to show a strong association?

3. If A is correlated with B, does this mean that A causes B?

The Multiplication Rule for Independent Events

• Two events A and B are independent if the occurence of A does not affect the probability that B occurs.

• Example 1:   Flip a coin twice. Define the event A to be getting a head on the first toss. Define the event B to be getting a head on the second toss. The events A and B are independent.

• The Multiplication Rule:   If A and B are independent, then P(A and B) = P(A) × P(B).

• Example 2:   Flip a coin 10 times. What is the probability of getting all heads?
  Ans: P(H and H and ... and H (10 heads)) = P(H) × P(H) × ... × P(H) = 0.5¹⁰ = 0.0009766.

• Example 3:   The probability that I will die in a single plane ride is 0.0001. What is my probability of dying in 500 independent plane crashes?
  Ans: Let A_i be the event of not dying in the ith plane flight. Then P(A₁ and ... and A₅₀₀) = P(A₁) × ... × P(A₅₀₀) = 0.9999⁵⁰⁰ = 0.951227. Therefore the probabability of dying in one of the 500 flights is 1 - 0.9512 = 0.04877.

• Example 4:   Roll a pair of dice 24 times. What is the probability of obtaining snakeeyes (pair of ones) at least once?

Errors in Medical Tests

• Today we look at how false positives and false negatives in testing can give misleading results.

• Not all tumors in the body are cancerous.
  Tumors caused by cancer are called malignant.

  Tumors not caused by cancer are called benign.

• Preliminary mammogram screening is 85% accurate.

• Here is an mammogram x-ray with a tumor outlined in red. Accurate identification of tumors as malignant or benign is not easy.

• If a woman tests positive in prelimary screening is she likely to have cancer? Not necessarily.

• Consider a study where mammograms are given to 10,000 women with tumors.

• In this population only 1% of breast tumors are cancerous (100 women have cancer, 9900 women have benign tumors).

• Preliminary mammogram screening correctly identifies 85% of the 100 (total of 85) malignant tumors as malignant. These are called true positives.

• The other 15% (total of 15) receive negative results, even though they actually have cancer. These are called false negatives.

• Preliminary mammogram screening correctly identifies 85% of the 9900 (total of 8415) benign tumors as benign. These are called true negatives.

• The other 15% of the 9900 (total of 1485) get positive results, which incorrectly indicate that their tumor is malignant. These are called false positives.

• Here are the results:

  	Tumor is Malignant	Tumor is Benign	Total
  Positive Mammogram	85 True Positives	1485 False Positives	1570
  Negative Mammogram	15 False Negatives	8415 True Negatives	8430
  Total	100	9900	10,000

• The total number of positive results is 1570, but only 85 are true positives, so the probability that a positive result means cancer is 85 / 1570 = 0.054 = 5.4%.

• The total number of negative results is 8430, but only 15 are false negatives, so the probability that a patient gets a negative result but has cancer anyway is 15/8430 = 0.0018 = 0.018%.

• Conclusion: the probability of cancer, given a positive test, depends not only on the characteristics of the test but also on the probability of having a malignant tumor in the first place.

Activity A9

• Work on Activity A9.