Probability: outcomes, tables, and mutually exclusive and independent events

Last time

Before we start: last time, in Time series, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.

How likely?

Probability measures how likely an event is, on a scale from 0, impossible, to 1, certain. This lesson covers listing outcomes, and probabilities from tables and charts, and the rules for mutually exclusive and independent events.

The basics

When outcomes are equally likely, the probability of an event is the number of favourable outcomes over the total number of outcomes. The probability that an event does not happen is 1 minus the probability that it does. Write probabilities as fractions, decimals or percentages, not as ratios.

Listing outcomes

Listing every outcome systematically, in a table or a list, makes sure none is missed. Two dice give 6 times 6, which is 36, equally likely outcomes; six of them have a total of 7. So P(total of 7) is 6 over 36, which is one sixth. Three dice give 6 cubed: 216 outcomes.

Probability from a table or chart

From a table or a bar chart, the probability is the count for the event divided by the right total. In a table of 50 students, 20 like tennis, 8 of them girls. P(a student likes tennis) is 20 over 50, which is 0.4. Given that a student likes tennis, the probability that she is a girl is 8 over 20, also 0.4: the total is now just the tennis column.

Mutually exclusive events

Mutually exclusive events cannot both happen, such as scoring a 2 and a 5 on one throw of a die. For them, P(A or B) is P(A) plus P(B), and P(A and B) is 0. If events can overlap, subtract the overlap: P(A or B) is P(A) plus P(B) minus P(A and B).

Independent events

Independent events do not affect each other, such as throwing a die and tossing a coin. For them, P(A and B) is P(A) times P(B). To test for independence, check whether P(A and B) equals P(A) times P(B).

A worked example

A and B are independent, with P(A) equal to 0.5 and P(B) equal to 0.2. P(A and B) is 0.5 times 0.2, which is 0.1. P(A or B) is 0.5 plus 0.2 minus 0.1, which is 0.6. P(neither) is 1 minus 0.6, which is 0.4. P(exactly one) is P(A or B) minus P(both): 0.6 minus 0.1, which is 0.5.

The rules

The probability rules in one table.

Mistakes that lose marks

Mistake one: adding the probabilities of events that can happen together, without subtracting the overlap. Mistake two: multiplying the probabilities of events that are not independent. Mistake three: using the whole table total when the question says given that. The fix: the right rule, and the right total.

The full-marks tip

The full-marks tip: name the rule before the numbers, such as: independent, so multiply. For a given-that question, write the restricted total first.

On one card

So, on one card: probability. Keep it for revision.