Chosen from everyone
A club has 24 members. 7 play music only, 5 play music and sport, 8 play sport only, and 4 play neither. Check: 7 + 5 + 8 + 4 = 24.
Pick a member at random from all 24. The members who play sport are the 5 in the overlap and the 8 in sport alone, so the probability that the member plays sport is .
The 24 members: 7 play music only, 5 both, 8 sport only, and 4 neither.
Given that a member plays music
Now the member is picked from those who play music. Everyone outside the music circle is ruled out, so only 7 + 5 = 12 members are left, and 5 of them also play sport. The probability that the member plays sport, given that they play music, is .
This is a conditional probability: a chance worked out when something is already known. Knowing one event happened cuts down the outcomes that are still possible, and the whole becomes the members in that event. Here the chance of sport changed from to , which is .
Given music, only the shaded circle counts: 7 + 5 = 12 members, and the 5 in the overlap play sport.
The condition decides the whole
Turn the question round: given that a member plays sport, what is the probability that they play music? Now the whole is the sport circle, 5 + 8 = 13 members, and 5 of them play music, so the answer is .
The top is the same 5 people in the overlap both times. The bottom is whichever group is given: 12 for music, 13 for sport.
Given sport, only the shaded circle counts: 5 + 8 = 13 members, and the same 5 in the overlap play music.
A smaller whole
Another set of 20 equally likely outcomes has 8 in event A, 10 in event B, and 6 in both. Out of all 20, . Given B, the whole shrinks to B's 10 outcomes, and 6 of them are in A, so the chance of A given B is .
P(A) = 8/20 counts A out of all 20 outcomes; conditioning on B shrinks the universe to B's 10
Condition on B and watch the denominator change
Slide the handle to condition on B: the box of 20 outcomes fades and B grows to fill the frame. The overlap stays 6, but the whole it is counted against changes from 20 to 10.
A rare illness and a test
In a town, 4 people in every 100 have an illness. A test says yes to 3 of every 4 people who are ill, and it wrongly says yes to 1 in 8 people who are well. Picture 100 people: 4 are ill and 96 are well. The test says yes to 3 of the 4 ill people and to 96 ÷ 8 = 12 of the well ones.
The test gives the right answer for 3 ill people and 84 well people, 87 of the 100, so it is right most of the time. But 3 + 12 = 15 people get a yes.
Each cell is one of 100 people. The 4 ill people are marked with a dot, and the 15 colored cells are the people the test says yes to: 3 ill and 12 well.
Given that the test said yes
Someone gets a yes. Given that, only the 15 people with a yes count, and 3 of them are ill. So the probability that a person with a yes is ill is .
Most of the yes results belong to well people. The test is wrong for only 1 in 8 well people, but there are 96 well people and only 4 ill ones, so the wrong yeses outnumber the right ones, 12 to 3.
The usual mistakes
Dividing by everyone: . Given music, the members who do not play music cannot be picked, so the whole is the 12 in the music circle.
Dividing by the music-only members: . The 5 in the overlap play music as well, so they belong in the whole: 7 + 5 = 12.
Turning the fraction upside down: . A probability is never more than 1; the members who play sport go on top, out of the 12.
Swapping what is given. The chance of sport given music is ; the chance of music given sport is . For the test, the chance of a yes given ill is , but the chance of ill given a yes is only .
Music and drama
In the application below, part (b) picks a student from those who take music. The whole is the 27 music students, not the 60 in the year group, and the students in the overlap who also take drama go on top.
Worked example: Students Taking Music and Drama in a Venn Diagram, and a Probability Read from One Region
Question In a year group of 60 Secondary 4 students, 27 take music, 21 take drama and 8 take both. (a) Draw a Venn diagram with the number of students in every region, and find the probability that a student chosen at random takes exactly one of the two subjects. (b) A student is chosen at random from those who take music. Find the probability that this student also takes drama.
1.Write the 8 students who take both subjects in the overlap of the two circles.
The 8 students who take both subjects go in the overlap first. 2.The music circle holds 27 students, so 27 − 8 = 19 take music only. The drama circle holds 21, so 21 − 8 = 13 take drama only.
Take the overlap off each circle: 27 − 8 = 19 take music only and 21 − 8 = 13 take drama only. 3.The circles hold 19 + 8 + 13 = 40 students, so 60 − 40 = 20 take neither subject. Check: 19 + 8 + 13 + 20 = 60.
60 − 19 − 8 − 13 = 20 students take neither, and the four regions add up to 60. 4.(a) Exactly one subject means music only or drama only, which is 19 + 13 = 32 students. The probability is 3260 = 815.
(a) Exactly one subject is 19 + 13 = 32 students, so the probability is 3260 = 815. 5.(b) The student is chosen from the music circle, so the whole is 27, not 60. 8 of those 27 students also take drama, so the probability is 827.
(b) Chosen from the music circle, the whole is 27, and 8 of them take drama: 827.
Answer: (a) the regions hold 19 music only, 8 both, 13 drama only and 20 neither, and the probability is 815; (b) 827
Common mistakes
- Writing 27 and 21 in the parts of the circles outside the overlap. Those totals already include the 8 students who take both, so the circles would hold 27 + 8 + 21 = 56 students instead of 40, and the 8 would be counted three times.
- Answering (b) with 860. That is the probability that a student from the whole year group takes both subjects. In (b) the student is already known to take music, so only the 27 students in the music circle can be chosen.
More probability with several events problems, worked step by step →