1. Complements and Conditional Probability
Section 4-5
Multiplication Rule:
Complements and Conditional Probability

2. Key Concepts
Probability of “at least one”: Find the probability that among several trials, we get at least one of some specified event.
Conditional probability: Find the probability of an event when we have additional information that some other event has already occurred.

3. Complements: The Probability of “At Least One”
“At least one” is equivalent to “one or more.”
The complement of getting at least one item of a particular type is that you get no items of that type.

4. Finding the Probability of “At Least One”
To find the probability of at least one of something, calculate the probability of none, then subtract that result from 1. That is,
P(at least one) = 1 – P(none).

5. Conditional Probability
A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred. P(B|A) denotes the conditional probability of event B occurring, given that event A has already occurred, and it can be found by dividing the probability of events A and B both occurring by the probability of event A:

6. Intuitive Approach to Conditional Probability
The conditional probability of B given A can be found by assuming that event A has occurred, and then calculating the probability that event B will occur.

7. Confusion of the Inverse
To incorrectly believe that P(A|B) and P(B|A) are the same, or to incorrectly use one value for the other, is often called confusion of the inverse.

8. Example 1: Provide a written description of the complement of the given event. When the 15 players on the L.A. Lakers basketball team are tested for steroids, at least one of them tests positive.
All 15 players test negative.

9. Example 2: Provide a written description of the complement of the given event. When four males are tested for a particular X-linked recessive gene, none of them are found to have the gene.
At least one of the four males has the X-linked recessive gene.

10. Example 3: If a couple plans to have six children, what is the probability that they will have at least one girl? Is that probability high enough for the couple to be very confident that they will get at least one girl in the six children?
1 – (.5)6 =
Yes

11. Example 4: If you make guesses for four multiple-choice test questions (each with five possible answers), what is the probability of getting at least one correct? If a very lenient instructor that says that passing the test occurs if there is at least one correct answer, can you reasonably expect to pass by guessing.
1 – (0.8)4 =
There is a good chance of passing with guesses, but it isn’t reasonable to expect to pass with guesses.

12. Example 5: Find the probability of a couple having a baby girl when their fourth child is born, given that the first three children were all girls. Is the result the same as the probability of getting four girls among four children?
0.5.
No.

13. Example 6: An experiment with fruit flies involves one parent with normal wings and one parent with vestigial wings. When this parents have an offspring, there is a ¾ probability that the offspring has normal wings and a ¼ probability of vestigial wings. If the parents give birth to 10 offspring, what is the probability that at least 1 of the offspring has vestigial wings? If researchers need at least one offspring with vestigial wings, can they be reasonably confident of getting one?
1 – (0.75)10 = 0.944
Yes, the probability is quite high, so they can be reasonably confident of getting at least one offspring with vestigial wings.

14. Example 7: Use the information in the table below to answer the following question. Assume that 1 of the 98 test subjects is randomly selected. Find the probability of selecting a subject with a positive test result, given that the subject did not lie. Why is this particular case problematic for test subjects?
Results from Experiments with Polygraph Instruments
Did the subjects actually lie?
No (did Not Lie)
Yes (Lied)
Positive test result 15 42
(Polygraph test indicated that the subject lied.)
(false positive)
(true positive)
Negative test result 32 9
(Polygraph test indicated that the subject did not lie.)
(true negative)
(false negative)

15. Of those 47 people, only 15 of them had positive test results.
Results from Experiments with Polygraph Instruments
Did the subjects actually lie?
No (did Not Lie)
Yes (Lied)
Positive test result 15 42
(Polygraph test indicated that the subject lied.)
(false positive)
(true positive)
Negative test result 32 9
(Polygraph test indicated that the subject did not lie.)
(true negative)
(false negative)
Knowing that they did not lie means that we will only look at those people.
Of those 47 people, only 15 of them had positive test results.
Therefore, the probability is:

16. Example 8: Use the data in the table below
Example 8: Use the data in the table below. Instead of summarizing observed results, the entries reflect the actual probabilities based on births of twins. Identical twins come from a single egg that splits into two embryos, and fraternal twins are from separate fertilized eggs. The table entries reflect the principle that among sets of twins, 1/3 are identical and 2/3 are fraternal. Also, identical twins must be of the same sex and the sexes are equally likely (approximately), and sexes of fraternal twins are equally likely.
a) After having a sonogram, a pregnant woman learns that she will have twins. What is the probability that she will have identical twins?
Sexes of Twins
boy/boy
boy/girl
girl/boy
girl/girl
Identical Twins 5
Fraternal Twins

17. Example 8 continued: Use the data in the table below
Example 8 continued: Use the data in the table below. Instead of summarizing observed results, the entries reflect the actual probabilities based on births of twins. Identical twins come from a single egg that splits into two embryos, and fraternal twins are from separate fertilized eggs. The table entries reflect the principle that among sets of twins, 1/3 are identical and 2/3 are fraternal. Also, identical twins must be of the same sex and the sexes are equally likely (approximately), and sexes of fraternal twins are equally likely.
b) After studying the sonogram more closely, the physician tells the pregnant woman that she will give birth to twin boys. What is the probability that she will have identical twins? That is, find the probability of identical twins given that the twins consist of two boys.
Sexes of Twins
boy/boy
boy/girl
girl/boy
girl/girl
Identical Twins 5
Fraternal Twins

18. Example 8 continued: Use the data in the table below
Example 8 continued: Use the data in the table below. Instead of summarizing observed results, the entries reflect the actual probabilities based on births of twins. Identical twins come from a single egg that splits into two embryos, and fraternal twins are from separate fertilized eggs. The table entries reflect the principle that among sets of twins, 1/3 are identical and 2/3 are fraternal. Also, identical twins must be of the same sex and the sexes are equally likely (approximately), and sexes of fraternal twins are equally likely.
c) If a pregnant woman is told that she will give birth to fraternal twins, what is the probability that she will have one child of each sex?
Sexes of Twins
boy/boy
boy/girl
girl/boy
girl/girl
Identical Twins 5
Fraternal Twins

19. Example 9: When testing blood samples for HIV infections, the procedure can be made more efficient and less expensive by combining samples of blood specimens. If samples from three people are combined and the mixture tests negative, we know that all three individual samples are negative. Find the probability of a positive result for three samples combined into one mixture, assuming the probability of an individual blood sample testing positive is 0.1 (the probability for the “at-risk” population, based on data from the New York State Health Department).