0580 · 8.1Introduction to probability

Probability scale and notation

0:00A bag contains eight blue counters and two yellow counters. Choosing blue is more likely than choosing yellow, but neither colour is guaranteed. Probability gives us a precise way to describe that uncertainty. In this video, you will learn the probability scale and calculate single event probabilities. You will also see why the probability of an event not happening is one minus the probability that it happens. You need only a little fraction knowledge to begin.

0:24A fraction compares a part with a whole, so three fifths means three equal parts out of five. Decimals and percentages can describe that same proportion. In probability, these different forms describe the same chance, rather than different answers. Probability measures how likely an event is to occur. An event is the outcome, or collection of outcomes, that we are interested in. For example, rolling an even number is an event.

0:48It includes several possible results, but we can still calculate one probability for the event. The probability scale runs from zero to one. Zero means impossible, while one means certain. A probability of one half means the event is equally likely to happen or not happen. Values closer to one describe more likely events, while values closer to zero describe less likely events. Suppose we roll an ordinary die numbered from one to six.

1:13Rolling seven is impossible because seven is not an available result. Rolling a number less than seven is certain because every available result satisfies that description. Rolling six is possible, but it is not certain. A probability cannot be negative or greater than one. Zero stands for no chance at all, and one stands for complete certainty. A percentage above one hundred per cent would claim more than certainty. If your answer lies outside the scale, check the information and your calculation.

1:41Probabilities can be written as fractions, decimals or percentages. One half, zero point five and fifty per cent all describe the same chance. The percentage scale runs from zero per cent to one hundred per cent. Unless a question specifies a form, any of these equivalent forms can express a probability. We use a capital letter to name an event. Let R mean rolling an even number. The notation P of R means the probability of that event.

2:06The letter inside the brackets identifies the event, while the P tells us that we are describing its probability. A small prime mark after the event letter means not that event. We read the probability notation as P of R prime, or the probability of not R. For our die, not rolling an even number means rolling an odd number. The prime mark changes the event being described, rather than changing the meaning of probability.

Counting equally likely outcomes

2:31To calculate a probability by counting, we need equally likely outcomes. This means each individual outcome has the same chance of being selected. For a fair die, each of its six faces has an equal chance. We can therefore compare the number of wanted outcomes with the total number of possible outcomes. Favourable outcomes means the outcomes that satisfy the event, not necessarily outcomes that are pleasant or desirable. The numerator counts those wanted outcomes.

2:56The denominator counts every possible outcome. Dividing gives the fraction of the complete set that belongs to our event. For our first worked example, a fair die is rolled once. Calculate the probability of rolling a number greater than four, and call this event E. The possible results are one, two, three, four, five and six. Only five and six satisfy the event, so there are two favourable outcomes. We divide the two favourable outcomes by the six possible outcomes.

3:25This gives 2/6. Dividing the numerator and denominator by two simplifies the answer to one third. Keep that exact fraction rather than replacing it with a rounded decimal. The equal chance condition matters because counting alone does not always measure chance. A spinner could have two labelled regions of very different sizes. Counting one region out of two would then ignore how much of the spinner it occupies. Only use the counting formula when the outcomes being counted are equally likely.

3:52Our second example uses information from a table. A bag contains six red counters, nine blue counters and five green counters. One counter is selected at random, with each individual counter equally likely to be chosen. Calculate the probability that it is blue, and call this event B. First, add six, nine and five to obtain twenty counters altogether. There are nine favourable counters, so P of B = 9/20.

4:18Multiplying the numerator and denominator by five gives 45/100. This is zero point four five, or forty-five per cent. Notice that there are three colours, but the probability of blue is not one third. We are choosing individual counters, and the colours contain different numbers of them. The total in the denominator must therefore be twenty counters, not three colour categories. Always identify exactly what is being selected. An examination could show the same information in a bar chart instead of a table.

4:48Read each bar using the vertical scale before calculating. The heights still give six red, nine blue and five green counters. The probability method stays unchanged because the graph presents the same counts in a different form.

Complements and not events

5:00The rule for an event not happening comes from splitting every possible result into two groups. Either the event happens, or it does not happen. These possibilities cover everything, and they cannot both occur on the same selection. Their probabilities must therefore add to one. The event not R is called the complement of R. Together, an event and its complement account for the whole probability. If we know the probability of the event, subtracting it from one leaves the probability of its complement.

5:26This explains the rule rather than making it an isolated formula to memorise. For our third worked example, we are told that P of B is zero point eight. Calculate P of B prime. The whole probability is one, so subtract zero point eight from one. The result is zero point two, meaning the probability of not B is twenty per cent. Check the answer by adding zero point eight and zero point two.

5:49Their sum is one, which confirms that the two probabilities account for every possibility. Subtracting from one works with fractions as well as decimals. If using percentages throughout, subtract the event percentage from one hundred per cent instead. Our fourth example shows twenty students in a Venn diagram. The circle represents event M, meaning the selected student studies music. Seven students are inside the circle, and thirteen are outside it.

6:14One student is chosen at random, and we must calculate the probability that this student does not study music. Adding seven and thirteen confirms that the total is twenty. The thirteen students outside the circle satisfy not M, so the probability is 13/20. Alternatively, subtract 7/20 from one. Writing one as 20/20 gives the same answer of 13/20.

Common mistakes and recap

6:37Common mistakes include leaving outcomes out of the denominator and confusing likely with certain. A high probability still allows the event not to happen. Another mistake is treating not blue as one particular alternative colour. In our counter bag, not blue includes both red and green, because both fail the condition blue. For that bag, P of B prime is 1 − 9/20, which is 11/20. These counting and complement methods should work without a calculator on Paper two.

7:04You can also use them on Paper four, where a calculator is available. Exact fractions usually avoid unnecessary rounding. When a decimal answer needs rounding, follow the question, normally using three significant figures unless instructed otherwise. Probability describes chance on a scale from zero to one. For equally likely outcomes, calculate the favourable number divided by the total number. Probability notation identifies the event, and a prime mark identifies its complement.

7:31Since an event and its complement cover every possibility, the probability of not happening equals one minus the probability of happening.