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### 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/
```
Associated Topics:
High School Probability

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