Probability

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Year 10 Interactive Maths - Second Edition

Probability

Recall that:

The sample space (S) of an experiment is the set of all possible outcomes of any trial of the experiment to be conducted.

An event (E) is a subset of the sample space. That is, an event is a subset of all possible outcomes. We refer to this subset of outcomes as favourable outcomes.

The probability of event E occurring is given by

This is often written as:

This result holds only if the outcomes of an experiment are equally likely.

Note:

The events are denoted by capital letters A, B, C, D, E, ...

Example 2

A die is rolled. Find:

a. the sample space for this experiment
b. the probability of obtaining an even number
c. the probability of obtaining a prime number

Solution:

A die.

(b) Let A be the event that an even number is obtained. Therefore, A = {2, 4, 6}. Pr(A) = n(A) / n(S) = 3 / 6 = 1 / 2

Pr(B) = n(B) / n(S) = 3 / 6 = 1 / 2

Range of Probability

If an event is impossible, its probability is 0. If an event is certain to occur, its probability is 1. The probability of any other event is between these two values. That is:

Pr(impossible event) = 0, Pr(certain event) = 1 and if A is any event, then 0 <= Pr(A) <= 1.

The probabilities for an impossible event, even chance and a certain event are 0, 1 / 2 and 1 respectively.

Example 3

A die is rolled. Find the probability of obtaining:
a. a 7
b. a number less than or equal to 6

Solution:

(b) Let A be the event that a number less than or equal to 6 is obtained. Then A = {1,2,3,4,5,6}. Now, Pr(A) = Number of outcomes in A / Number of outcomes in S = 6 / 6 = 1

Note:

It is certain that event A will occur as it contains all 6 possible outcomes.
7 is not an outcome of rolling a die as it is not possible.

Example 4

A pack of 52 playing cards consists of four suits, i.e. clubs, spades, diamonds and hearts. Each suit has 13 cards which are the 2, 3, 4, 5, 6, 7, 8, 9, 10, jack, queen, king and the ace card. Clubs and spades are of black colour whereas diamonds and hearts are of red colour. So, there are 26 red cards and 26 black cards.

Find the probability of drawing from a well-shuffled pack of cards:
a. a black card
b. the king of diamonds
c. a jack

Solution:

(a) A pack of 52 cards has 26 black cards. So, Pr(a black card) = 26/52 = 1/2

(b) A pack of 52 cards has 1 king of diamonds. So, Pr(the king of diamonds) = 1 / 52

Complement of Event A

A^'is the complement of event A. It contains all of the elements in the sample space S that are not included in A.

A Venn diagram depicting the event A and its complement A'.

It is certain that either A or A^' must occur. So, it follows that:

For any event A and its complement A^':

Example 5

Solution:

Let A be the event that a train will be late. Then A^' is the event that it will be on time.

Pr(A) + Pr(A') = 1 so Pr(A') = 99/100

Key Terms

favourable outcomes, range of probability, impossible, certain, complement

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