Poker, the Lottery

Date: 11/30/98 at 12:19:40
From: mike
Subject: Probability

(1) What is the probability that a 5-card poker hand contains cards of
    5 different values?

(2) 6 integers are selected at random from the set (1,2,...,40). What 
    is the probability that none of the integers 1,...,6 will be 
    selected?

(3) To play the Pennsylvania superlottery, a player selects 7 of the
    first 80 positive integers. What is the probability that the
    person will win the grand prize by picking 7 integers that are 
    among the 11 integers selected randomly by the Pennsylvania 
    lottery commission?

(4) 6 integers are chosen at random from the set of the first 40 
    positive integers. What is the probability that your 6 choices 
    match exactly 5 of these integers?

(5) Which is more likely: rolling a total of exactly 8 using two fair
    dice or rolling a total of exactly 8 when three fair dice are 
    rolled?

Date: 12/01/98 at 10:11:26
From: Doctor Anthony
Subject: Re: Probability

>(1) What is the probability that a 5-card poker hand will contain 
>cards of 5 different values?

[5 different face values, not in sequence, not all cards in the same 
suit]                     
        
There are   13C5 x 4^5  ways with all cards of different face value.  
There are 4C1 x 13C5 ways where cards are all of the same suit, and 
10 x 4 ways where cards are in sequence.

So Prob(5 different face values, not in sequence, not in same suit)

             13C5 x 4^5 - 4C1 x 13C5 - 4 x 10
           = ---------------------------------   =  0.50509
                         52C5


>(2) 6 integers are selected at random from the set (1,2,...,40). 
>What is the probability that none of the integers 1,...,6 will be selected?

  You must select 6 from 34 numbers.  This can be done in C(34,6) ways
 
                         C(34,6) 
Required probability =  --------  =  0.35038
                         C(40,6)

>(3) To play the Pennsylvania superlottery, a player selects 7 of the
>first 80 positive integers. What is the probability that the
>person wins the grand prize by picking 7 integers that are among
>the 11 integers selected randomly by the Pennsylvania lottery
>commission?
 
                   C(11,7)
   Probability  = ---------  = 3.149 x 10^(-11)
                   C(80,11)

>(4)  6 integers are chosen at random from the set of the first 40 
>positive integers. What is the probability that your 6 choices match 
>exactly 5 of these integers?

                         C(6,5) x C(34,1)
Required probability =  ------------------  =  5.3147 x 10^(-5)
                             C(40,6)

>(5) Which is more likely : rolling a total of exactly 8 using two 
>fair dice or rolling a total of exactly 8 when three fair dice are 
>rolled?

With two dice, possibilities are  6+2   Each row has probability 1/36
                                  5+3
                                  4+4   Total prob. =  5/36
                                  3+5
                                  2+6

With three dice   1+1+6  (3 of these)  Each row has probability  1/216
                  1+2+5  (6 of these)
                  1+3+4  (6 of these)   Total prob. =  21/216
                  2+2+4  (3 of these)
                  2+3+3  (3 of these)
                     ------------------
                  Total = 21 possibilities


Comparing the two  5/36 = 0.13888
                 21/216 = 0.09722

So the two dice have the greater probability.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/