# The A to Z of mathematics a basic guide - Sidebotham T.H.

ISBN 0-471-15045-2

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Probability of rolling a three = |

The probability of an event happening is another way of answering the question: “What are the chances or likelihood that an event will happen?” When we roll a die it has an equal chance of landing on any of the six numbers 1, 2, 3, 4, 5, or 6, provided it is not biased. It is certain to land on one of these numbers, but which one? The chances, or probability, of it landing on the number 3 is one chance in six. If you rolled the die six times, in theory, it is expected to land once on a 3 and once on each of the other five numbers. In practice, however, in six rolls the die may not land on a 3 at all, or it could land on a 3 every time! All we can say is this: If we roll a die, the probability of it landing on a 3 is |, provided each of the numbers on the die are equally likely to turn up. Suppose you rolled a die four times and it landed on a 6 every time. The next time you roll the die the probability of it landing on a 6 is still |. But the probability of rolling five 6’s in succession is (|) = 1/7776. When we calculate probabilities we write them as fractions, or may express them as decimals or percentages.

Example 1. A die is rolled once. What is the probability the outcome is an odd number?

Solution. The number of ways of rolling an odd number is three, because the die could land as a 1, 3, or 5. The total number of outcomes is six, because the die could land as a 1, 2, 3, 4, 5, or 6. Therefore,

Probability of rolling an odd number = |

= | Canceling down the fraction

Example 2. In a bag there are five red marbles, three green and two blue. If John puts his hand in the bag and, without looking, brings out a marble, what is the probability that the marble is:

(a) Red?

(b) Blue?

344 PROBABILITY OF AN EVENT

(c) Red or green?

(d) Red and green?

Solution. There are 10 marbles in the bag and 5 of them are red. Write

number of ways of choosing a red marble

Probability of choosing a red marble = ---------------------------—-----:---------——

total number of ways of choosing a marble

_ _5_

10

= | Canceling down the fraction

(b) There are 10 marbles in the bag and 2 of them are blue. Write

Probability of choosing a blue marble = ^ Using the formula for probability

= | Canceling down the fraction

(c) There are 10 marbles in the bag and 8 are either red or green. Write

Probability of either a red or green =

= | Canceling down the fraction.

(d) It is impossible to have marbles in the bag that are both red and green. Write

Probability of a red and green = ^

= 0

We have seen in the example above that if the probability of an event is zero, then that event is impossible. In addition, if an event is certain to happen, its probability is one. An example of an event that is certain to happen is that every human being will die. The probability that we will die is one.

All probabilities lie between zero and one, inclusive.

There are three kinds of probabilities we now need to study:

♦ The probability of mutually exclusive events

♦ The probability of complementary events

♦ The probability of independent events

Before proceeding it may be a good idea to read the entry Complementary Events, where all these terms are explained.

The probability of mutually exclusive events are explained in the first example.

PROBABILITY OF AN EVENT 345

Example 3. Ann rolls a die once.

(a) What is the probability she rolls a 3 and a 6?

(b) What is the probability she rolls a 3 or a 6?

Solution, (a) When one die is rolled, the event of rolling a 3 and the event of rolling a 6 are events that cannot both happen at the same time, and are called mutually exclusive events. So the probability of rolling a 3 and a 6 is impossible on one roll of a die, and equal to zero.

(b) The probability of rolling a 3 or a 6 is also a mutually exclusive event and is calculated by adding the probability of each event:

Probability of rolling a3or6=| + | Prob(3) + prob(6).

= | Canceling the fraction.

Summary. If E\ and £2 are mutually exclusive events, then

Probability of E\ and £'2 = 0 Probability of £1 or £2 = probability of £1 + probability of £2

The probability of complementary events are explained in the next example.

Example 4. Using the same example of Helen rolling a die once, what is the probability she rolls an even number or an odd number?

Solution. The event of rolling an even number and the event of rolling an odd number are mutually exclusive events, because they both cannot happen at the same time, so we add the probabilities. In addition, these two events make up all the possible outcomes, so they are complementary events. Write

Probability of an even number or odd number = | + | Prob(even) + prob(odd)

= 1

For complementary events the sum of the probabilities is always 1.

Summary. If £1 and £2 are complementary events, then

Probability of £1 or £2 = probability of £1 + probability of £2 = 1

When the outcome of one event has no effect on the outcome of another event, we say that the two events are independent events. To obtain the probability of independent events we multiply the probabilities of the separate events.

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