AMS 311, Lecture 10

February 27, 2001

Review of problems for examination

  1. Let A, B, and C be three events. Show that exactly two of these events will occur with probability

The event that exactly two of A, B, and C occur is ABCC ABCC ACBC.

Each of the three sets is disjoint.

By Theorem 1.5, Then,

  1. Let A be an event such that P(A)=0.5, and let B be an event such that P(B)=0.8. What is the maximum possible value of P(AB) and the minimum possible value of P(AB)?

Recall The maximum of

The maximum of P(AB) is the minimum of P(A) and P(B), so that the minimum of P(AB)

is the larger of P(A) and P(B). The maximum is 1.0; the minimum is 0.8.

  1. AThe Great Carsoni,@ a magician, claims to have extrasensory perception. In order to test this claim, he is asked to identify the 4 red cards out of 4 red and 4 black cards which are laid face down on the table. AThe Great Carsoni@ correctly identifies 3 of the 4 red cards. Thereafter, he claims to have proved his point. What is the probability that AThe Great Carsoni@ would have correctly identified exactly 3 of the 4 red cards if he were, in fact, guessing?

The Great Carsoni chooses 3 of 4 red cards and 1 of 4 black cards with probability

  1. A fair die is rolled 7 independent times. What is the probability that at least one face will not appear?

The probability of the sequence 1123456 is There are 6 choices for the duplicate face, and there are choices of positions for the 1 face, 2 face,  The probability that all faces will appear is and the probability that at least one face will not appear is .

  1. A box contains 8 good and 2 defective items. Four items are selected at random without replacement from the box. What is the probability of finding at least one of the defective items?

Finding the probability of at least one defective is more easily found by using the complement of no defectives found:

  1. The probability that a randomly selected individual has a specified medical condition is .05. A screening test for this condition has sensitivity .99. That is, the probability that an affected individual tests positive is .99. The specificity of the procedure is .95. That is, the probability that an individual not affected tests negative is .95. What is the probability that the screening test will produce a false positive reading? That is, given that the results of the test are positive, what is the probability that the individual does not have the condition?

This is an application of Bayes’ Theorem. Let H be the event the subject is healthy, and S the event the subject is affected. Let + be the event that the test is positive. The probability of a false positive is P(H|+). This is I often make errors in interpreting the problem. Make sure that you are solving the problem requested. Note that

  1. In poker, a flush consists of 5 cards of the same suit. Given that all of the cards in a randomly selected poker hand (that is, a draw of 5 cards) are red (either hearts or diamonds), what is the conditional probability that the hand is a flush?

The probability that the five cards drawn are all red is The probability of a heart flush or a diamond flush given all red cards drawn is

  1. Each day an experimental animal is exposed to a certain set of stimuli designed to elicit a particular response. Let Ak be the event that the animal makes the desired response on the kth day, and suppose that P(Ak+1|Ak)=β and that P(Ak+1|Akc)=α, where 0<α<β1. Let pk=P(Ak).

Show that pk+1=α+(β-α)pk.

Use the law of total probability: Here, Grouping the terms with given the result.