Introduction
Probability is the likelihood or chance of an event occurring.
Probability
= the number of ways of achieving success the total number of
possible outcomes
For example, the probability of flipping a coin and it
being heads is ½, because there is 1 way of getting a head and the total number
of possible outcomes is 2 (a head or tail). We write P(heads) = ½ .
The
probability of something which is certain to happen is 1. The probability of
something which is impossible to happen is 0. The probability of something
not happening is 1 minus the probability that it will happen.
Single Events:
Example
There are 6 beads in a bag, 3
are red, 2 are yellow and 1 is blue. What is the probability of picking a
yellow?
The probability is the number of yellows in the bag divided by
the total number of balls, i.e. 2/6 = 1/3.
Example
There is a bag full of
coloured balls, red, blue, green and orange. Balls are picked out and replaced.
John did this 1000 times and obtained the following results: Number of blue
balls picked out: 300 Number of red balls: 200 Number of green balls:
450 Number of orange balls: 50
a) What is the probability of picking a
green ball? b) If there are 100 balls in the bag, how many of them are likely
to be green?
a) For every 1000 balls picked out, 450 are green. Therefore
P(green) = 450/1000 = 0.45
b) The experiment suggests that 450 out of
1000 balls are green. Therefore, out of 100 balls, 45 are green (using
ratios).
Multiple Events:
Possibility Spaces
When working out what the probability of two things happening is, a
probability/ possibility space can be drawn. For example, if you throw two dice,
what is the probability that you will get: a) 8, b) 9, c) either 8 or 9?
a) The black blobs indicate the ways of getting 8 (a 2 and a
6, a 3 and a 5, ...). There are 5 different ways. The probability space shows us
that when throwing 2 dice, there are 36 different possibilities (36 squares).
With 5 of these possibilities, you will get 8. Therefore P(8) = 5/36 . b)
The red blobs indicate the ways of getting 9. There are four ways, therefore
P(9) = 4/36 = 1/9. c) You will get an 8 or 9 in any of the 'blobbed'
squares. There are 9 altogether, so P(8 or 9) = 9/36 = 1/4 .
Probability Trees
Another way of representing 2 or more events is on a
probability tree.
Example
There are 3 balls in a bag: red, yellow and blue. One ball is
picked out, and not replaced, and then another ball is picked out.
The first ball can be red, yellow or blue. The probability is
1/3 for each of these. If a red ball is picked out, there will be two balls
left, a yellow and blue. The probability the second ball will be yellow is 1/2
and the probability the second ball will be blue is 1/2. The same logic can be
applied to the cases of when a yellow or blue ball is picked out
first.
In this example, the question states that the ball is not
replaced. If it was, the probability of picking a red ball (etc.) the second
time will be the same as the first (i.e. 1/3).
The AND and OR rules
In the above example, the probability of picking a red first is 1/3 and a
yellow second is 1/2. The probability that a red AND then a yellow will be
picked is 1/3 × 1/2 = 1/6 (this is shown at the end of the branch). The
probability of picking a red OR yellow first is 1/3 + 1/3 = 2/3. When the
word 'and' is used we multiply. When 'or' is used, we add. On a probability
tree, when moving from left to right we multiply and when moving down we
add.
Example
What is the probability of getting a yellow and a red in any
order? This is the same as: what is the probability of getting a yellow AND a
red OR a red AND a yellow. P(yellow and red) = 1/3 × 1/2 = 1/6 P(red and
yellow) = 1/3 × 1/2 = 1/6 P(yellow and red or red and yellow) = 1/6 + 1/6 =
1/3
Copyright © Matthew Pinkney 2003
