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Wize University Statistics Textbook > Counting Techniques

Counting and Probability

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Counting and Probability

Combinations calculate the number of outcomes where we do not care about the order. To calculate the probability of a combination, we must take into consideration of the number of favorable outcomes over the number of possible outcomes.

Example
There are 26 cases, and 5 of them contains the word "GOLD". In order to win a million dollars, the contestant must select, without replacement, 3 cases that contains the word "GOLD". What is the probability that they will win a million dollars?

Number of favorable outcomes:

5C3=10_{5}C_{3}=10 (there are 10 different ways to choose 3 of 5 "GOLD" cases)

Number of possible outcomes:

26C3=2600_{26}C_{3}=2600 (there are 2,600 different ways to choose 3 different cases out of 26 cases)

Probability of choosing 3 "GOLD" cases:

= [Number of favorable outcomes] divided by [Number of possible outcomes]

=102600=0.0038=\frac{10}{2600}=0.0038


Wize Tip
You can also solve this using Hypergeometric Probability.

Example: Counting and Probability

There are 5 male employees and 3 female employees in the marketing department. What is the probability of randomly selecting 2 of each to launch a new campaign?

You need to select 2 out of 5 male employees AND (multiplication rule) select 2 out of 3 female employees → You are selecting 4 out of 8 total employee.


(5C2 )⋅(3C2)(8C4)\displaystyle{\frac{\left(_{5}C_{2}\ \right)\cdot\left(_3{C}_{2}\right)}{\left(_{8}C_{4}\right)}}


(10)(3)70=3070=0.4286\frac{\left(10\right)\left(3\right)}{70}=\frac{30}{70}=0.4286

Translation: there are 30 different ways of selecting 2 of each gender out of 70 different ways to select 4 of 8 employees.
Seven people are right-handed, and 3 people are left-handed. What is the probability of randomly selecting 2 of each?
Extra Practice
Counting and Probability
In how many ways can we rearrange the letters in the word SEVEN to form a unique string of letters?
Counting and Probability
In a group of 10 people, 6 are right-handed and 4 are left-handed. If we choose 4 people at random without replacement, what is the probability that exactly 2 of them are right-handed? Give your answer as a decimal.
Counting and Probability
Six people are right-handed, and 4 people are left-handed. What is the probability of randomly selecting 2 of each?
Probability
A library has 20 math textbooks, 10 French textbooks, and 15 geography textbooks. Cassie checked out 5 textbooks.
[Enter both answers with at least 2 decimal places, e.g. 0.26]
Counting and Probability
Six people are right-handed, and 4 people are left-handed. What is the probability of randomly selecting 2 of each?
Probability
There are 20 DVD’s in the clearance bin at Target, and 40% of them are comedy and 60% of them are dramas.
Counting and Probability
You draw 3 cards from a deck of 52 cards without replacement. Determine the probability that all three of them are of the same colour.
Counting and Probability
You draw 3 cards from a deck of 52 cards without replacement. Determine the probability that none of the cards are hearts.
Probability
Parts (i) to (vi) pertain to the following information:
In order to win a prize, you must flip 'Heads' at least 2 times out of 3 trials. However, the coin is not fair because there is a 40% probability you will flip 'Heads' each time (and therefore 60% probability you will flop 'Tails' each time).
(i) Complete the probability distribution table. Enter 3 decimal places for probabilities.
Probability
(ii) What is the probability that you will win a prize?