38.7. GENERAL CONSIDERATIONS PROBABILITY 779

If X is a random variable having values in {0,1, · · · ,k} such that the probability thatX = j is given by the jth term of the above sum, then X is said to have a hyperge-ometric distribution. Much much more can be said about this topic. Hint: If youpick k things from N things m of which are marked and N−m unmarked, there arevarious ways to do it determined by the value of j, the number of marked things outof your sample of k things.

19. Suppose a pair of dice has one blue and the other one red. What is the probabilitythat when they are rolled the blue die delivers a strictly larger number than the reddie? Now what is the probability that either this happened or a 6 is rolled? What isthe probability that the blue is greater than the red and a 6 is rolled?

20. Recall the following table illustrating the possible outcomes of rolling a pair of dice.

1,1 1,2 1,3 1,4 1,5 1,62,1 2,2 2,3 2,4 2,5 2,63,1 3,2 3,3 3,4 3,5 3,64,1 4,2 4,3 4,4 4,5 4,65,1 5,2 5,3 5,4 5,5 5,66,1 6,2 6,3 6,4 6,5 6,6

Find the probability that you roll a 7 before you roll either a 3 or an 11. Hint: It canhappen in infinitely many distinct ways. You don’t roll either a 3 or an 11 for k rollsand then on the kth roll you get a 7. Here k = 0,1,2,3, · · · so you need to take a limitof the partial sums associated with the different values of k and then take a limit. Sowhat is the probability of getting a 7 on try k+1 and not getting either a 3 or an 11before this? Argue it is

( 1318

)k 16 .

21. Explain why in general, P(A∪B) = P(A) +P(B)−P(A∩B) where A,B are twoevents such as in the above problem having the blue be larger than the red die orrolling a 6.

22. Let X be the random variable which gives the number of heads when you flip a coin6 times. Which value of X has the highest probability? What is the expected valueof X? What is the variance of X . For these last parts, see Problem 13 and 14 above.

23. You have a class of 12 students who will be seated in four rows consisting of threestudents in each row. Jeroboam, Nadab, Baasha, and Elah must sit in the front for dis-ciplinary reasons. Also, you absolutely must not have Baasha sitting next to Nadabbecause Baasha is a thug who will attack Nadab. If Baasha is to sit on the front leftseat as viewed by the teacher, what is the probability that an acceptable outcome willoccur if the students take their seats completely at random?

38.7 General Considerations ProbabilityAs mentioned earlier, X is a random variable if a probability is associated with X beingfound in some set of possible values. The following examples have been discussed either inthe chapter or in the exercises. These are examples of discrete random variables because the