Probability

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?

For every 1000 balls picked out, 450 are green. Therefore P(green) = 450/1000 = 0.45

b) If there are 100 balls in the bag, how many of them are likely to be green?

The experiment suggests that 450 out of 1000 balls are green. Therefore, out of 100 balls, 45 are green (using ratios).

Multiple Events

Independent and Dependent Events

Suppose now we consider the probability of 2 events happening. For example, we might throw 2 dice and consider the probability that both are 6's.

We call two events independent if the outcome of one of the events doesn't affect the outcome of another. For example, if we throw two dice, the probability of getting a 6 on the second die is the same, no matter what we get with the first one. It's still 1/6.