This note introduces the language of probability: experiments, outcomes, events and sample spaces. You will calculate probabilities of simple events, use complementary events, work with Venn diagrams and sample space diagrams, and find the expected number of occurrences.
Key ideas
Vocabulary.
- An experiment (or trial) is any process with an uncertain result, such as rolling a die.
- An outcome is one possible result, such as rolling a 4.
- The sample space, often written \(U\), is the set of all possible outcomes. For one die, \(U=\{1,2,3,4,5,6\}\).
- An event is a set of outcomes we are interested in, such as "an even number", \(A=\{2,4,6\}\).
Probability of an event. If all outcomes are equally likely, then \[ P(A)=\frac{n(A)}{n(U)}, \] where \(n(A)\) is the number of outcomes in the event and \(n(U)\) is the total number of outcomes. Probabilities satisfy \(0\le P(A)\le1\). An impossible event has probability 0 and a certain event has probability 1. The probabilities of all the outcomes add to 1.
Complementary events. The complement \(A'\) is the event "\(A\) does not happen". Then \[ P(A')=1-P(A). \] This is very useful for "at least one" problems: \(P(\text{at least one})=1-P(\text{none})\).
Relative frequency. If a trial is repeated many times, the proportion of times an event occurs is its relative frequency, which gets close to the probability as the number of trials grows. It can be used to estimate probabilities when outcomes are not equally likely.
Expected number of occurrences. If an event has probability \(P\) in each trial and there are \(n\) trials, the expected number of occurrences is \[ n\times P. \] This is an average over many repetitions, not a guaranteed result.
Reading the wording. "Not \(A\)" means the complement. "\(A\) and \(B\)" means both events happen, which is the overlap of the circles. "\(A\) or \(B\)" means at least one of them happens. In a Venn diagram, "only \(A\)" means the part of circle \(A\) that is outside circle \(B\). Always decide which region you need before you count.
Diagrams.
- Sample space diagram: a table or grid listing all outcomes of two events, for example two dice. Count the cells that match the event.
- Venn diagram: a rectangle for \(U\) with overlapping circles for events. The overlap is \(A\cap B\) ("\(A\) and \(B\)"), and everything in at least one circle is \(A\cup B\) ("\(A\) or \(B\)"). Fill in the overlap first, then work outwards, and put the number outside all circles last.
- Tree diagram: branches show successive events, with probabilities written on the branches. Probabilities on branches from one point add to 1.
Definitions and formulae
| Term or formula | Meaning / when to use |
|---|---|
| \(P(A)=\dfrac{n(A)}{n(U)}\) | Probability when all outcomes are equally likely. |
| \(0\le P(A)\le1\) | Every probability lies between 0 and 1. |
| \(P(A')=1-P(A)\) | Probability that \(A\) does not happen. |
| \(A\cap B\) | Both \(A\) and \(B\) happen (intersection). |
| \(A\cup B\) | \(A\) or \(B\) or both happen (union). |
| Expected occurrences \(=nP\) | Average number of times an event occurs in \(n\) trials. |
| Relative frequency | \(\dfrac{\text{times the event occurred}}{\text{number of trials}}\); estimates probability. |
Worked examples
Example 1 Two fair dice are rolled and the scores are added. (a) Find \(P(\text{sum}=8)\). (b) Find \(P(\text{sum}\ge10)\). (c) Find \(P(\text{at least one six})\). (d) The dice are rolled 120 times. How many times is a sum of 7 expected?
The sample space diagram has \(6\times6=36\) equally likely outcomes.
(a) Sum 8: \((2,6),(3,5),(4,4),(5,3),(6,2)\), which is 5 outcomes. \(P=\dfrac{5}{36}\).
(b) Sum 10: \((4,6),(5,5),(6,4)\) is 3. Sum 11: \((5,6),(6,5)\) is 2. Sum 12: \((6,6)\) is 1. Total 6, so \(P=\dfrac{6}{36}=\dfrac16\).
(c) No six on either die: \(5\times5=25\) outcomes. So \(P(\text{at least one six})=1-\dfrac{25}{36}=\dfrac{11}{36}\).
(d) Sum 7: \((1,6),(2,5),(3,4),(4,3),(5,2),(6,1)\) is 6 outcomes, so \(P=\dfrac{6}{36}=\dfrac16\). Expected number \(=120\times\dfrac16=20\).
Answer: (a) \(\tfrac5{36}\); (b) \(\tfrac16\); (c) \(\tfrac{11}{36}\); (d) 20 times
Example 2 In a group of 40 students, 22 play football (\(F\)), 15 play tennis (\(T\)) and 6 play both. Draw the Venn diagram numbers and find (a) \(P(F\text{ only})\), (b) \(P(\text{neither sport})\), (c) \(P(F')\).
Both: 6. Football only: \(22-6=16\). Tennis only: \(15-6=9\).
In at least one circle: \(16+6+9=31\). Outside both: \(40-31=9\).
(a) \(P(F\text{ only})=\dfrac{16}{40}=0.4\).
(b) \(P(\text{neither})=\dfrac{9}{40}=0.225\).
(c) \(P(F')=1-\dfrac{22}{40}=\dfrac{18}{40}=0.45\).
Answer: (a) 0.4; (b) 0.225; (c) 0.45
Example 3 Three fair coins are tossed. (a) List the sample space. (b) Find \(P(\text{exactly two heads})\) and \(P(\text{at least one head})\). (c) The experiment is repeated 200 times. How many times is "exactly two heads" expected?
(a) HHH, HHT, HTH, THH, HTT, THT, TTH, TTT. That is 8 equally likely outcomes.
(b) Exactly two heads: HHT, HTH, THH, which is 3. So \(P=\dfrac38\).
At least one head is the complement of TTT: \(1-\dfrac18=\dfrac78\).
(c) \(200\times\dfrac38=75\).
Answer: (b) \(\tfrac38\) and \(\tfrac78\); (c) 75 times
Common errors
- Common error: using \(P=\dfrac{n(A)}{n(U)}\) when outcomes are not equally likely, such as treating "sum of two dice" as 11 equally likely outcomes (2 to 12). List all 36 pairs instead.
- Common error: treating \((2,5)\) and \((5,2)\) as the same outcome for two distinguishable dice. They are different outcomes.
- Common error: filling a Venn diagram by writing the totals (22 and 15) straight into the circles. Subtract the overlap first.
- Common error: giving a probability above 1 or a negative one. Always check that your answer is between 0 and 1.
- Common error: rounding an expected number to a whole number too early or saying it is guaranteed. It is an average, so give the value and call it expected.
Exam tip: For "at least one" questions, work out the probability of "none" and subtract it from 1. It is nearly always shorter than adding up all the cases.
Check yourself
- A spinner numbered 1 to 4 is spun and a coin is tossed. How many outcomes are in the sample space?
Answer
\(4\times2=8\) outcomes, for example (1,H), (1,T), (2,H), and so on. - A number is chosen at random from 1 to 20. Find the probability that it is not prime.
Answer
Primes: 2, 3, 5, 7, 11, 13, 17, 19, which is 8 numbers. Not prime: \(20-8=12\). \(P=\dfrac{12}{20}=\dfrac35\). - In a set of 30 items, \(n(A)=12\), \(n(B)=10\) and \(n(A\cap B)=4\). Find the probability that an item is in neither \(A\) nor \(B\).
Answer
\(n(A\cup B)=12+10-4=18\). Neither: \(30-18=12\). \(P=\dfrac{12}{30}=0.4\). - The probability of rain on a given day is 0.15. How many rainy days are expected in 200 days?
Answer
\(200\times0.15=30\) days.