Bill likes to play a game in which he shuffles the

Bill likes to play a game in which he shuffles the deck, turns over 4 cards, and
looks for pairs of cards that have the same value. What is the chance that Bill finds at least one
pair of cards that have the same value?

1 – P(No Pairs)

Two ways to calculate P(No Pairs)
13

o Select each card one at a time: (12/12)(10/11)(8/10)(6/9) = 16/33
o Use combinations:
 6
C
4
= Number of ways to choose 4 numerical values

2 = Number of cards at each value
 12
C
4
= Number of ways to choose 4 cards from 12

(
6
C
4
* 2 * 2 * 2 * 2)/
12
C
4
= 16/33
o 1 – 16/33 = 17/33

Alternative method for calculating the probability of No Pairs (i.e., assume you draw
serially and find the probability for each of the four draws given the “no pairs”
restriction—i.e., on the first draw, all 12 cards are candidates, on the second draw, all
remaining 11 cards are candidates except 1 card, etc.):
o (12/12) * (10/11) * (8/10) * (6/9) = 16/33
Note: The problems above can also be done by drawing the cards serially and multiplying
the individual probabilities together.
SET THEORY
Use a Double Set Matrix (i.e., use a 3-by-3 grid)

Good when you have two attributes, each with two values

Third row/column is for "Total"

E.g., persons at a party are (i) either M or F and (ii) either Smart or Stupid
AuB = A + B – (AnB)
If there are three sets A, B, and C, then :

Total number of people/Number of people in at least one set: 5
O AuBuC = A + B + C – AnB – AnC – BnC + AnBnC

Number of people in exactly one set:
O A + B + C – 2*AnB – 2*AnC – 2*BnC + 3*AnBnC

Number of people in exactly two of the sets"
O AnB + AnC+ BnC – 3*AnBnC

Number of people in two or three sets:
O AnB + AnC + BnC – 2*AnBnC

Number of people in exactly three of the sets =
O AnBnC
STATISTICS
A = S/n
 Average (Arithmetic Mean) = (sum of all numbers)/(total # of numbers)
Standard Deviation:
5
This is the only formula worth remembering.
14

 Must be greater or equal to zero.
 3 ways to express:
o
Sqrt(mean squared distances of the numbers from the mean of the numbers)
o
Sqrt(mean of the squares of the numbers minus squared mean of the numbers)
o
Sqrt(variance of the set)
Complex Weighted Average Problems
 Consider plugging in numbers to save time!
 Use ratios to solve if you are given the following, (i) the average of quantities X and Y,
(ii) the value/weight/measure of X and (iii) the value/weight/measure of Y.
o
Step 1: Find the distance of X from the average = D
X
o
Step 2: Find the distance of Y from the average = D
Y
o
Step 3: Step up an inverse propotion: X:Y = D
Y
:D
X
 Example: An island is populated by only Fatties and Skinny people. Skinny people
weigh on average 5 pounds less than the average weight of islanders. The Fatties weight
15 pounds more than the on average weight of islanders.
o
What is the ratio of Skinny people to Fatties? S:F = 15:5 = 3:1
 Explanation: You ratio of Skinny people to Fatties is inversely
proportional to their weight (i.e., you need 3 Skinny people to balance out
each Fatty).