Question

  In games of the KENO or LOTTO style, the bettor selects numbers from a fixed set. Then the game operator selects another set of numbers, and the bettor wins according to the number of matches.   

a.​Suppose that the game uses the numbers 1 through 50, and suppose that the operator selects eight of these. If the bettor selects five numbers, find the probability that there are exactly five matches. HINT: Think Hypergeometric

b.​Suppose that the game uses the numbers 1 through 50, and suppose that the operator selects ten of these. If the bettor selects five numbers, find the probability that there are exactly five matches. Also note whether this probability is larger or smaller than the probability in a.   

c.​Suppose that the game uses the numbers 1 through 50, and suppose that the operator selects ten of these. If the bettor selects six numbers, find the probability that there are exactly six matches. Note whether the probability here is larger or smaller than the probability in b.

Answer 1

Let us assume that the better is going first.

Formula for hyper-geometric distribution is :

Where r=Number of success in the population

x= Number of successes in the sample

N= Population size

n=Sample size

PART a)

There are two ways of approaching this question:

1) Using hyper geometric distribution: N=50, r=5 [ 5 successes defined by bettor] , n=8 [ a sample of 8 drawn by the operator] and x=5 [ The number of matches for success needed]

2) Using calculation: The bettor creates 5 successes and 45 failures. When game operator selects 8 numbers, there are ways of make that selection. the number of ways the bettor can select five matching number (success) and select three non-matching number is . Thus probability is :   

=

You can also think of the operator going first. Thus using the hyper geometric distribution, N=50, r=8 [8 successes defined by the operator ] , n=5 [ a sample of 5 drawn by bettor] and x=5 [ The number of matches needed for success]

PART b)

Way of selecting 10 is . The number of ways in which operator can select five matching and five non matching choices are . Thus the probability is :

Thus you can observe that probability is higher than Part a).

Again, one can use hyper geometric distribution; then N=50, r=10 [10 successes as described by the operator], n=5 [a sample of 5 drawn by the bettor] and x=5 [the number of matches needed for success].

PART c)

Way of selecting 10 is . The number of ways in which bettor can select six matching number and four non matching number is : . Thus the probability is :

Comparing that with Part b) we can see that probability in part c) is less that in b) , the probability is almost 1/9 th times of Part a). This implies ,getting five of five matches is easier that getting six-of-six matches.