SAT – Probability

Q:  What is the probability of randomly guessing the last four digits of a telephone number correctly?

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A: 1/10,000

Explanation:  Each digit is chosen from the integers 0 through 9, which is ten possible choices for each digit.  There are four digits to choose independently, so there are:

10 x 10 x 10 x 10 = 10,000 ways to choose the four digits.

Then, the one correct sequence of 4 digits, is one out of 10,000.  The probability is 1/10,000.

Another way to arrive at the number 10,000 is as follows:

The 4-digit sequence may be anything from 0000 to 9999.  So, how many is that?  9999 – 0000 +1 = 10,000.  Why the ‘plus one’ on the end?  If you just subtract the numbers, it tells you how many spaces there are between the numbers.  Add one to include either the first number or the last number, whichever one has not been already counted.

 

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SAT – Counting and Ordering

Q: There are 11 basketball teams in a tournament. Each team plays each other team once. How many games are played?

 Answer: 55

Explanation 1: This question is really ‘How many ways are there to choose 2 teams from a total of 11 teams, if the order of the teams is not important?’ It’s a combination question. Find the nCr function on your calculator. On the TI-84 Plus it’s in MATH-PRB-3. Type in 11 nCr 2. It’s 55. (There is a longer way to do it by hand, using the formula for nCr.)

Explanation 2: Team A plays one game against each of the 10 other teams. That’s 10 games so far. Team B plays 9 other games (its game against Team A has already been counted). Team C plays 8 other games, … all the way down to the 1 last game between teams J and K. Add: 10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 55.

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SAT – Counting and Ordering

Q:  20 students are to choose a president and vice president from among themselves. How many possible ways can they be chosen?

Answer:  380

Explanation 1:  Using the Counting Principle, set up two blanks to represent the president and vice president, from left to right:

____    ____

There are 20 choices for the president.  Put 20 in the first blank:

_20_    ____

The same person cannot be chosen for both positions, so there are 19 left to choose from for the vice president.  Put 19 in the second blank:

_20_    _19_

Now, multiply these numbers.  20 x 19 = 380.

Explanation 2:  This is a permutation problem since you are choosing 2 people from a total of 20 people, and the order is important.  The order is important because the two people chosen must be given specific positions, and switching those positions results in a different scenario.  Do  20 nPr 2 , or use the formula for nPr.  On the TI-84 Plus, find nPr in MATH-PRB-2.  The result is 380.

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SAT – Counting and Ordering

Q: In how many ways can the letters of the word ‘FIVE’ be arranged?

Answer:   24

Explanation 1:  SInce there are 4 different letters, do 4!  (that’s ‘4 factorial’).   4! = 4 x 3x 2x 1 = 24.  Your calculator might have a factorial (!) function.  On the TI-84 Plus, find it in MATH – PRB.

Explanation 2:  SInce you are choosing 4 items from a total of 4 items, and the order is important, you can do 4 nPr 4, or use the formula for nPr.

 

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SAT: Counting and Ordering

Q: How many ways are there to go from point A to point B on the rectangular prism below, by traveling only along exactly 3 edges of the prism?

10620806_820546241324048_5097600483628304775_n

Answer: 6

Explanation: There are 3 ways to travel away from A, shown below in green. There are 2 ways to continue along each of these paths to B, shown in blue. That’s 3 x 2 = 6 paths to B along three edges of the prism. This is easier than making a mess of the diagram trying to draw all 6 paths (and making it hard to count them).

10561771_820546701324002_1707217643831692413_n

 

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SAT – Counting and Ordering

Q:  When 5 people met for dinner, each person shook hands with each other person once.  How many handshakes happened?

Answer: 10

Explanation 1: Two people = a handshake.  This question is really ‘How many ways are there to choose 2 people from a total of 5, if the order is not important?’ It’s a combination question. Find the nCr function on your calculator. Type in 5 nCr 2. It’s 10. (There is a longer way to do it by hand, using the formula for nCr.)

Explanation 2: Person A shook hands with 4 people.  That’s  4 handshakes so far. Person B shook hands with 3 other people (his shake with A has already been counted). Person C shook hands with 2 other people.  Person D shook hands with 1 other person (E). Add: 4 + 3 + 2 + 1 = 10.

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SAT – Counting and Ordering

Q: Four people are to stand side by side for a photo. The shortest two people must stand at the ends, while the tallest two people must stand in the center. How many ways are there to arrange the people for the photo?
 
Answer: 4
Explanation: Use the Counting Principle.  Set up four blanks:
___  ___  ___  ___ to represent the four positions.
Decide on the number of possibilites for each position.
The first position has 2 possibilities (the 2 shortest people):
_2_  ___  ___  ___.  Whichever of the two shortest people isn’t in the first spot, will have to be in the fourth spot:
_2_  ___  ___  _1_.
Do the same for the middle two positions, using the two tallest people:
_2_  _2_  _1_  _1_.  Now, multiply these numbers together: 4.
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