WESTERN CAPE EDUCATION DEPARTMENTAlison Kitto 082 802 3295
GRADE 12 PROBABILITY
Important concepts covered up to the end of Grade 11 include the following:
The identity P(A or B) = P(A) + P(B)P(A and B) for any events A and B.
The addition law for mutually exclusive events P(A or B) = P(A) + P(B) (Since P(A and B) =0)
The multiplication law for independent events P(A and B) = P(A) P(B)
[The need for events to be independent before multiplying the individual probabilities together must be stressed. Ignoring this leads to errors!]
The subtraction law for complementary events (events which are mutually exclusive and exhaustive)
P(A) = 1P(not A). This law is often useful when a question requires the probability of “at least one” item being chosen.
Venn diagrams
Tree diagrams
Two way contingency tables
Example 1
Consider the menu at Fredah’s Fast Foods:
Main courseDessertDrink
Curry and riceIce creamOrange juice
Pap and beansFruit saladMilk
Boerewors rollPumpkin frittersTea
ChickenPancakesCoffee
Vegetarian saladCola
Lemonade
In how many different ways can a person make up an order for a main course, followed by a dessert and a drink?
Example 2
How many different standard old style Gauteng number plates were possible if the format was 3 alphabetic characters (excluding Q and the vowels A, E, I, O and U) followed by 3 numeric digits? BBB000 is one option, KLM789 is another. BAT123 is not an option as it contains an A, which is one of the letters excluded.
The new style Gauteng numberplates are: two alphabetic characters, again excluding Q and A, E, I, O, U), then two numeric digits and then two more alphabetic characters. How many more number plates can be accommodated in the new style than was possible with the 3 alphabetic and 3 numeric characters?
Example 3
A certain maze is constructed for a mouse so that at exactly 8 places, the mouse
has to make a choice between turning left or right. How many routes are there through the maze?
Example 4
If Nkosi has 10 shirts, 5 pairs of trousers and 3 pairs of shoes, how many different outfits can he make up using one of each item of clothing?
Example 5
The employees in a manufacturing company are required to compile codes to punch into a computer at the entrance to the factory where they are employed. The code must be 4 characters long and can be any combination of the 26 letters of the alphabet and the 10 numerical digits: 0 to 9. How many different codes are possible?
Example 6
Use the fundamental counting principle to determine the answers to the following questions. Try to describe how each example differs from the previous examples discussed.
6.1In how many different ways can different stamps be placed in a row in the top right hand corner of an envelope if there are:
6.1.13 different stamps
6.1.24 different stamps
6.1.35 different stamps
6.1.4mdifferent stamps?
6.2Consider the number of different ways an electronic timing device in a swimming pool can identify positions in a race. Ignore the possibility of any swimmers touching at exactly the same time. How many different results are possible if we require the first:
6.2.1three positions in a 10 lane pool;
Hint: In major races, the fastest swimmers are placed in the middle lanes, so 5, 4, 6 might be a likely order, but if all possible orders are to be considered, we must be methodical: 1,2,3 or 1,2,4 or 1,2,5 or … or 1,2,10. Then say 1,3,2, or 1,3,4 or 1,3,5 or … or 1,3,10 and so on.
6.2.2four places in an eight lane pool;
6.2.3five places in a twelve lane pool;
6.2.4mplaces in an n lane pool?
Example 7
These are the flags of the countries that competed in the African Nations Cup in 2006.
7.1In how many ways could the flags have been displayed in a single row?
7.2In how many different ways could the winner and runner up have been
determined, assuming that every nation could win and every other nation could be
the runner up?
Example 8:
How many code numbers of three digits can be made using the digits 1, 2, 3, 4 and 5 if the order of the digits is important and:
8.1repetitionis not permitted?8.2 repetition is permitted?
Example 9
9.1Write down all the different four letter arrangements or ‘words’ that don’t need to have any meaning, that can be formed using the following letters:
9.1.1LIKE9.1.2LEEK9.1.3LULL9.1.4LULU
9.2Use your own examples to investigate longer words where one or more lettersmay be repeated. How many ‘words’ or different arrangement are possible if we start with 5 letters of which there are 3 of one letter and 2 of another? What if there are 4 of one and the 5th letter is different? What if there are 2 of one letter 2 of a second letter and the 5th letter is different?
Example 10
In a musical scale, there are seven basic notes, as described in the classic movie “The Sound of Music”. They are doh, re, me, fah, so, la and ti. How many different 7 note ‘tunes’ can be created if:
10.1any note may be used once only10.2a note may be used more than once
10.3notes may be repeated and the ‘tune’
must start and end with doh?
10.4the ‘tune’ must have three ‘doh’s, two
‘me’s and two ‘so’s /
Example 11
Using all the letters of the word NDUNDULU
11.1how many different ‘words’ can be formed?
11.2how many of the ‘words’ in 11.1 start with N?
11.3how many of the ‘words’ in 11.1 start and end with N?
Exercise 12
In how many different ways can the letters of the following names be arranged:
12.1LEBO12.2PETE12.3NASEEM
12.4HENNIE12.5NTLOTLENG12.6ANNAMARIE?
Using the fundamental counting principle in probability problems
Example 13
Suppose we take all the letters in the word HISTORY and arrange them in any order, without repetition. What is the probability that the ‘word’ will start with the ‘H’ and end with the ‘Y’?
Example 14:
Suppose that a number plate is formed using three letters of the alphabet, excluding the vowels and Q followed by any three digits. Calculate the probability that a number plate, chosen at random,
14.1starts with a ‘B’ and ends with a ‘5’
14.2has exactly one ‘B’
14.3has at least one ‘5’?
Example 15:
If the letters of the word STATISTICS are randomly arranged, what is the probability that the word will start and end with the same letter?
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