Probability of One Event
Probabilities Based on Equally Likely Events

When the outcomes of an event are all equally likely, then probabilities can be found by considering all the possible outcomes.

For example, when you toss a coin there are two possible outcomes, either heads or tails.

So

p(head)
p(tail)

The probability of an outcome is given by

provided all the outcomes are equally likely.

Worked Examples

1

A card is taken at random from a full pack of playing cards with no jokers. What is the probability that the card:

First note that each card is equally likely to be selected, and that there are 52 possible outcomes.
(a)

is an ace,

Show me

There are 4 aces, so

p(ace) =
=
(b)

is black,

Show me

There are 26 black cards, so

p(black) =
=
(c)

is a heart,

Show me

There are 13 hearts in the pack, so

p(heart) =
=
(d)

has an even number on it?

Show me

There are 20 cards with even numbers on them, so

p(even number) =
=
2

In a class of 30 students, 16 are girls, 4 wear glasses and 3 are left handed.

A student is chosen at random from the class. What is the probability that this student is:

All the students in the class are equally likely to be selected, when the choice is made at random.
(a)

a girl,

Show me

In the class there are 16 girls, so

p(girl) =
=
(b)

right-handed,

Show me

There are 3 left handed students and so the other 27 must be right handed.
So,

p(right handed) =
=
(c)

wearing glasses.

Show me

There are 4 students wearing glasses so,

p(wears glasses) =
=

Exercises

Robert takes a card at random from a full pack of 52 playing cards. What is the probability that his card:

(a)
is a diamond,
(b)
is a spade,
(c)
is a seven,
(d)
is a king,
(e)
has a prime number on it?

Repeat question 1, this time for a pack of playing cards containing 2 jokers (a total of 54 cards).

(a)
a diamond
(b)
a spade
(c)
a seven
(d)
a king
(e)
has a prime number on it

When you roll a die, what is the probability of obtaining:

(a)
a six,
(b)
a five,
(c)
an even number,
(d)
a prime number?

In a class of 32 students, 20 have school lunches and the rest bring their own food. What is the probability that a student chosen at random from the class brings their own food?

In a 'lucky dip' game at a school fair, a tub contains 50 prizes at the start of the fair. There are 20 superballs, 10 pens, 10 toy cars and 10 packets of sweets.

What is the probability that the first person to visit the lucky dip:

(a)
wins a superball,
(b)
does not win a pen,
(c)
wins a packet of sweets,
(d)
does not win a toy car.

If the first person wins a pen, what is the probability that the second person:

(e)
wins a pen,
(f)
does not win a toy car,
(g)
wins a packet of sweets?

Lester has the following 5 coins in his pocket:

£2,   £1,   50 p,   20 p,   10 p

He selects one coin at random to put in a charity collection box.

What is the probability that he:

(a)
gives more than £1,
(b)
has less than £2 left in his pocket,
(c)
has more than £3 left in his pocket,
(d)
gives away less than half the money in his pocket?

A bag contains 5 red counters, 3 green counters and 2 blue counters. Counters are taken out of the bag at random, but are not put back into the bag.

(a)

What is the probability that the first counter taken out is green?

(b)

If the first counter is green, what is the probability that the second counter is green?

(c)

If the first two counters are green, what is the probability that the third counter is green?

(d)

If a red counter is followed by a blue counter, what is the probability that the third counter is green?

In a raffle, 200 tickets are sold.

(a)

Holly buys one ticket. What is the probability that she wins first prize?

There are lots of prizes.

(b)

The probability that Holly wins a prize is . How many prizes are there?

Sophie is shown a selection of ski-holiday brochures. There are three for Italy, two for Austria and five for Switzerland.

Sophie takes one of these brochures at random.

(a)

What is the probability that it is for Italy?

(b)

What is the probability that it is not for Austria?

Challenge!

Anthony, Patrick and Millie repeatedly take turns tossing a die. Anthony begins; Patrick always follows Anthony; Millie always follows Patrick; Anthony always follows Millie.
What is the probability that Millie will be the first one to toss a six?

Investigation

1.
How many students must you gather together so as to ensure that at least two of the students have birthdays falling in the same month?
2.
A secondary school has an enrolment of 1100. Is it possible that there will be four students in the school whose birthdays fall on the same day of the year? Explain your answers.