Introduction to probability Revision Notes

Probability quantifies how likely an event is to happen, with values always between 0 and 1 inclusive. A probability of 0 means the event cannot happen, while a probability of 1 means the event is certain to happen. Values between 0 and 1 indicate varying degrees of likelihood. For example, a probability of 0.5 means the event is equally likely to happen or not happen.

Probabilities can be expressed in three common forms: fractions (e.g., 1/4), decimals (e.g., 0.25), or percentages (e.g., 25%). These forms are interchangeable and can be converted easily. For instance, 1/4 = 0.25 = 25%.

• Probability values range from 0 (impossible) to 1 (certain).
• Probabilities can be expressed as fractions, decimals, or percentages.
• Understanding the scale helps interpret how likely events are.

To find the probability of a single event, use the formula:

\[ \text{Probability of event} = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}} \]

The total number of possible outcomes must be known and all outcomes should be equally likely. For example, when rolling a fair six-sided die, the total number of outcomes is 6. The probability of rolling a 3 is therefore 1 favourable outcome divided by 6 possible outcomes, or \( \frac{1}{6} \).

• Identify all possible outcomes.
• Count the number of outcomes that satisfy the event.
• Divide the favourable outcomes by total outcomes to find probability.

Sometimes probabilities are not given directly but must be calculated from data presented in tables, bar charts, or Venn diagrams involving up to two sets. For example, a table might show the number of students who like football and basketball. To find the probability that a randomly chosen student likes football, divide the number of students who like football by the total number of students.

Venn diagrams help visualise overlapping sets. When dealing with two sets, the total number of outcomes is the total number of elements in the universal set. The number of favourable outcomes is the number of elements in the event set.

• Extract total and favourable counts from tables or graphs.
• Use Venn diagrams to identify elements in one or both sets.
• Calculate probability by dividing favourable outcomes by total outcomes.

The probability that an event does not happen is called the complement of the event. Since the event either happens or does not happen, their probabilities add up to 1. Therefore,

\[ P(\text{not } A) = 1 - P(A) \] where \( P(A) \) is the probability of the event occurring.

For example, if the probability that a counter is blue is 0.8, then the probability that it is not blue is \( 1 - 0.8 = 0.2 \).

• Understand that event and its complement probabilities sum to 1.
• Subtract the event probability from 1 to find the complement.
• Use this to find probabilities of 'not' events quickly.

A bag contains 5 red counters, 3 blue counters, and 2 green counters. A counter is drawn at random. Calculate:

1. The probability that the counter is blue, expressed as a fraction, decimal, and percentage.

2. The probability that the counter is not blue, expressed as a decimal and percentage.

• Total counters = 5 + 3 + 2 = 10.
• Probability blue = \( \frac{3}{10} = 0.3 = 30\% \).
• Probability not blue = \( 1 - 0.3 = 0.7 = 70\% \).