Probability

The One Thing

Theoretical probability is calculated by dividing the number of favourable outcomes by the total number of equally likely outcomes. Every probability lies between 0 and 1 and may be expressed as a fraction, decimal, or percentage.

Definitions and Results

• Experiment: A planned activity or process that produces a definite set of possible results.
• Outcome: A possible result of a random experiment.
• Sample Space: The set of all possible outcomes of an experiment, usually denoted by $S$.
• Event: A collection of one or more outcomes from the sample space.
• Favourable Outcomes: Outcomes that satisfy the condition of the event being considered.
• Equally Likely Outcomes: Outcomes that have the same chance of occurring, such as getting heads or tails with a fair coin.
• Theoretical Probability: The probability of an event calculated from the possible outcomes, without performing the experiment. It requires the outcomes to be equally likely.
• Probability of an Event: For an event $E$,

$$P(E)=\frac{\text{Number of favourable outcomes}}{\text{Total number of equally likely outcomes}}.$$ If $S$ is the sample space, $$P(E)=\frac{n(E)}{n(S)}.$$

• Range of Probability: For any event $E$,

$$0\le P(E)\le 1.$$ A probability closer to 1 indicates that an event is more likely; a probability closer to 0 indicates that it is less likely.

• Impossible Event: An event that cannot occur; its probability is

$$P(E)=0.$$

• Certain Event: An event that must occur; its probability is

$$P(E)=1.$$

• Complementary Event: The event in which the given event does not occur, written as $\text{not }E$ or $E^{\prime }$. Its probability is

$$P(\text{not }E)=1-P(E),$$ and $$P(E)+P(\text{not }E)=1.$$

• Fair Coin: The sample space is

$$S=\{H,T\}.$$ Since heads and tails are equally likely, $$P(H)=\frac{1}{2}.$$

• Fair Die: The sample space is

$$S=\{1,2,3,4,5,6\}.$$ The even outcomes are $2,4,6$, so $$P(\text{even number})=\frac{3}{6}=\frac{1}{2}.$$

• Standard Pack of Cards: When a card is drawn from a standard pack of 52 cards, each card is treated as equally likely if the pack is well shuffled.
• Percentage Form: A probability may be converted into a percentage by multiplying it by $100$.

Worked Methods

Calculating theoretical probability

• Identify the experiment and list its sample space.
• Confirm that the outcomes are equally likely.
• Identify the event and count its favourable outcomes.
• Count all outcomes in the sample space.
• Apply

$$P(E)=\frac{n(E)}{n(S)}.$$

• Express the result as a fraction, decimal, or percentage if required.

Example: Tossing a fair coin

• The sample space is

$$S=\{H,T\}.$$

• The event is obtaining a head.
• There is one favourable outcome, $H$.
• There are two total outcomes.
• Therefore,

$$P(H)=\frac{1}{2}.$$

Example: Rolling a fair die

• The sample space is

$$S=\{1,2,3,4,5,6\}.$$

• The event is obtaining an even number.
• The favourable outcomes are $2,4,6$, giving $n(E)=3$.
• The total number of outcomes is $n(S)=6$.
• Therefore,

$$P(\text{even number})=\frac{3}{6}=\frac{1}{2}.$$

Calculating a complementary probability

• Identify the probability of the event $E$.
• Identify its complement, written as $\text{not }E$ or $E^{\prime }$.
• Use

$$P(\text{not }E)=1-P(E).$$

• Alternatively, use

$$P(E)+P(\text{not }E)=1.$$

Converting probability to a percentage

• Find the probability as a fraction or decimal.
• Multiply it by $100$.
• State the result with the percentage sign.

For example, a probability of $\frac{1}{2}$ corresponds to $$\frac{1}{2}\times 100=50\%.$$

Where It Goes Wrong

• Applying the theoretical probability formula when the outcomes are not equally likely; the method requires equally likely outcomes.
• Omitting possible results from the sample space, so the total number of outcomes is incorrect.
• Counting favourable outcomes without following the exact condition stated in the event.
• Confusing an event with its complement; the complement is the event in which the given event does not occur.
• Forgetting that an impossible event has probability $0$ and a certain event has probability $1$.
• Giving a probability outside the required range $0\le P(E)\le 1$, or failing to multiply by $100$ when a percentage is requested.

What Gets Asked

• Define an experiment, outcome, sample space, event, favourable outcome, or theoretical probability.
• List the sample space for a fair coin or fair die.
• Calculate the probability of an event using

$$P(E)=\frac{n(E)}{n(S)}.$$

• Find the probability of obtaining a head with a fair coin.
• Find the probability of obtaining an even number with a fair die, including

$$\frac{3}{6}=\frac{1}{2}.$$

• Determine whether an event is impossible or certain.
• Calculate the probability of a complementary event using

$$P(\text{not }E)=1-P(E).$$

• Convert a probability into a percentage.
• Explain why theoretical probability depends on equally likely outcomes and a complete sample space.